PermPal Database

Av(1234, 3412)

Generating Function

\(\displaystyle -\frac{42 x^{12}-342 x^{11}+1336 x^{10}-3358 x^{9}+5775 x^{8}-7049 x^{7}+6231 x^{6}-4006 x^{5}+1856 x^{4}-604 x^{3}+131 x^{2}-17 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{4} \left(x -1\right)^{7}}\)

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Counting Sequence

1, 1, 2, 6, 22, 86, 333, 1235, 4339, 14443, 45770, 138988, 407134, 1157576, 3212157, ...

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Implicit Equation for the Generating Function

\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{4} \left(x -1\right)^{7} F \! \left(x \right)+42 x^{12}-342 x^{11}+1336 x^{10}-3358 x^{9}+5775 x^{8}-7049 x^{7}+6231 x^{6}-4006 x^{5}+1856 x^{4}-604 x^{3}+131 x^{2}-17 x +1 = 0\)

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Recurrence

\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 86\)
\(\displaystyle a \! \left(6\right) = 333\)
\(\displaystyle a \! \left(7\right) = 1235\)
\(\displaystyle a \! \left(8\right) = 4339\)
\(\displaystyle a \! \left(9\right) = 14443\)
\(\displaystyle a \! \left(10\right) = 45770\)
\(\displaystyle a \! \left(11\right) = 138988\)
\(\displaystyle a \! \left(12\right) = 407134\)
\(\displaystyle a \! \left(n +6\right) = -\frac{n^{6}}{180}+\frac{n^{5}}{10}-\frac{13 n^{4}}{72}-\frac{17 n^{3}}{12}+\frac{787 n^{2}}{360}-16 a \! \left(n \right)+80 a \! \left(n +1\right)-136 a \! \left(n +2\right)+112 a \! \left(n +3\right)-49 a \! \left(n +4\right)+11 a \! \left(n +5\right)-\frac{461 n}{60}+1, \quad n \geq 13\)

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Explicit Closed Form

\(\displaystyle \frac{\left(288 \sqrt{5}+720\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{360}+\frac{\left(-288 \sqrt{5}+720\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{360}+\frac{\left(15 n^{3}+45 n^{2}-3300 n +6480\right) 2^{n}}{360}+\frac{n^{6}}{180}+\frac{19 n^{4}}{72}+\frac{n^{3}}{4}+\frac{263 n^{2}}{360}-\frac{29 n}{4}-21\)

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Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).

This specification was found using the strategy pack "Insertion Point Col Placements" and has 261 rules.

Found on January 18, 2022.

Finding the specification took 93 seconds.

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This specification was found using the strategy pack "Insertion Point Row Placements" and has 278 rules.

Found on January 18, 2022.

Finding the specification took 105 seconds.

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This specification was found using the strategy pack "Insertion Row And Col Placements" and has 425 rules.

Found on January 18, 2022.

Finding the specification took 117 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right) F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right) F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{117}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{302}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{103}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{183}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{12}\! \left(x \right) F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{12}\! \left(x \right) F_{139}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{12}\! \left(x \right) F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{11}\! \left(x \right) F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{12} \left(x \right)^{2}\\ F_{161}\! \left(x \right) &= F_{12}\! \left(x \right) F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{11}\! \left(x \right) F_{165}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{11}\! \left(x \right) F_{162}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{170}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{25}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{11}\! \left(x \right) F_{153}\! \left(x \right) F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{175}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{178}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{176}\! \left(x \right)\\ F_{179}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{180}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{175}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{184}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{185}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)+F_{268}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{216}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{188}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{191}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{190}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{188}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{193}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{198}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{195}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{12}\! \left(x \right) F_{196}\! \left(x \right)\\ F_{196}\! \left(x \right) &= 2 F_{197}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)+F_{201}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{12}\! \left(x \right) F_{200}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{188}\! \left(x \right)+F_{202}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{12}\! \left(x \right) F_{203}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{205}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{206}\! \left(x \right)+F_{210}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{207}\! \left(x \right)+F_{208}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{11}\! \left(x \right) F_{209}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{203}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{12}\! \left(x \right) F_{211}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{212}\! \left(x \right)+F_{214}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{12}\! \left(x \right) F_{213}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{11}\! \left(x \right) F_{215}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{209}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{217}\! \left(x \right)+F_{219}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{218}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{220}\! \left(x \right)+F_{257}\! \left(x \right)\\ F_{220}\! \left(x \right) &= F_{221}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{222}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{230}\! \left(x \right)\\ F_{223}\! \left(x \right) &= F_{224}\! \left(x \right)+F_{226}\! \left(x \right)\\ F_{224}\! \left(x \right) &= F_{225}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{197}\! \left(x \right)\\ F_{226}\! \left(x \right) &= F_{227}\! \left(x \right)+F_{228}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{35}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{220}\! \left(x \right)+F_{229}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{12}\! \left(x \right) F_{231}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{232}\! \left(x \right)+F_{233}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{12}\! \left(x \right) F_{225}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{234}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{256}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{236}\! \left(x \right)+F_{255}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{237}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{238}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{239}\! \left(x \right)+F_{251}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{12}\! \left(x \right) F_{240}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{241}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{242}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{243}\! \left(x \right)+F_{245}\! \left(x \right)+F_{246}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{244}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{242}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{241}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{247}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{248}\! \left(x \right)\\ F_{248}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{246}\! \left(x \right)+F_{249}\! \left(x \right)\\ F_{249}\! \left(x \right) &= F_{250}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{250}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{248}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{252}\! \left(x \right)+F_{253}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{11}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{254}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{12}\! \left(x \right) F_{153}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{166}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{12}\! \left(x \right) F_{153}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{258}\! \left(x \right)+F_{260}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{259}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{257}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{261}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{261}\! \left(x \right) &= F_{262}\! \left(x \right)+F_{263}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{264}\! \left(x \right)+F_{265}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{19}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{265}\! \left(x \right) &= F_{257}\! \left(x \right)+F_{266}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{267}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{11}\! \left(x \right) F_{203}\! \left(x \right) F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{12}\! \left(x \right) F_{269}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{270}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{271}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{272}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{272}\! \left(x \right) &= F_{273}\! \left(x \right)+F_{299}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{274}\! \left(x \right)+F_{281}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{275}\! \left(x \right)+F_{276}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{277}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{11}\! \left(x \right) F_{278}\! \left(x \right)\\ F_{278}\! \left(x \right) &= F_{279}\! \left(x \right)+F_{280}\! \left(x \right)\\ F_{279}\! \left(x \right) &= F_{12} \left(x \right)^{2}\\ F_{280}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{281}\! \left(x \right) &= F_{282}\! \left(x \right)+F_{285}\! \left(x \right)\\ F_{282}\! \left(x \right) &= F_{283}\! \left(x \right)+F_{284}\! \left(x \right)\\ F_{283}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{284}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{285}\! \left(x \right) &= F_{286}\! \left(x \right)\\ F_{286}\! \left(x \right) &= F_{287}\! \left(x \right)+F_{296}\! \left(x \right)\\ F_{287}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{289}\! \left(x \right)\\ F_{288}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{289}\! \left(x \right) &= F_{270}\! \left(x \right)+F_{290}\! \left(x \right)\\ F_{290}\! \left(x \right) &= F_{12}\! \left(x \right) F_{291}\! \left(x \right)\\ F_{291}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{245}\! \left(x \right)+F_{292}\! \left(x \right)+F_{294}\! \left(x \right)\\ F_{292}\! \left(x \right) &= F_{293}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{293}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{291}\! \left(x \right)\\ F_{294}\! \left(x \right) &= F_{295}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{295}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{291}\! \left(x \right)\\ F_{296}\! \left(x \right) &= F_{12}\! \left(x \right) F_{297}\! \left(x \right)\\ F_{297}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{298}\! \left(x \right)\\ F_{298}\! \left(x \right) &= F_{11}\! \left(x \right) F_{242}\! \left(x \right)\\ F_{299}\! \left(x \right) &= F_{300}\! \left(x \right)+F_{301}\! \left(x \right)\\ F_{300}\! \left(x \right) &= F_{19}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{301}\! \left(x \right) &= F_{159}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{302}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{303}\! \left(x \right)\\ F_{303}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{304}\! \left(x \right)+F_{336}\! \left(x \right)\\ F_{304}\! \left(x \right) &= F_{305}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{305}\! \left(x \right) &= F_{306}\! \left(x \right)+F_{307}\! \left(x \right)\\ F_{306}\! \left(x \right) &= F_{303}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{307}\! \left(x \right) &= F_{308}\! \left(x \right)+F_{309}\! \left(x \right)\\ F_{308}\! \left(x \right) &= F_{218}\! \left(x \right)\\ F_{309}\! \left(x \right) &= F_{310}\! \left(x \right)\\ F_{310}\! \left(x \right) &= F_{311}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{311}\! \left(x \right) &= F_{312}\! \left(x \right)+F_{329}\! \left(x \right)\\ F_{312}\! \left(x \right) &= F_{307}\! \left(x \right)+F_{313}\! \left(x \right)\\ F_{313}\! \left(x \right) &= F_{12}\! \left(x \right) F_{314}\! \left(x \right)\\ F_{314}\! \left(x \right) &= F_{315}\! \left(x \right)+F_{326}\! \left(x \right)\\ F_{315}\! \left(x \right) &= F_{316}\! \left(x \right)\\ F_{316}\! \left(x \right) &= F_{12}\! \left(x \right) F_{317}\! \left(x \right)\\ F_{317}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{318}\! \left(x \right)\\ F_{318}\! \left(x \right) &= F_{319}\! \left(x \right)+F_{322}\! \left(x \right)\\ F_{319}\! \left(x \right) &= F_{320}\! \left(x \right)\\ F_{320}\! \left(x \right) &= F_{321}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{321}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{319}\! \left(x \right)\\ F_{322}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{323}\! \left(x \right)+F_{325}\! \left(x \right)\\ F_{323}\! \left(x \right) &= F_{324}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{324}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{322}\! \left(x \right)\\ F_{325}\! \left(x \right) &= F_{318}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{326}\! \left(x \right) &= F_{327}\! \left(x \right)+F_{328}\! \left(x \right)\\ F_{327}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{328}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{329}\! \left(x \right) &= F_{12}\! \left(x \right) F_{330}\! \left(x \right)\\ F_{330}\! \left(x \right) &= F_{331}\! \left(x \right)+F_{333}\! \left(x \right)\\ F_{331}\! \left(x \right) &= F_{332}\! \left(x \right)\\ F_{332}\! \left(x \right) &= F_{12}\! \left(x \right) F_{209}\! \left(x \right)\\ F_{333}\! \left(x \right) &= F_{334}\! \left(x \right)+F_{335}\! \left(x \right)\\ F_{334}\! \left(x \right) &= F_{12}\! \left(x \right) F_{327}\! \left(x \right)\\ F_{335}\! \left(x \right) &= F_{264}\! \left(x \right)+F_{266}\! \left(x \right)\\ F_{336}\! \left(x \right) &= F_{337}\! \left(x \right)\\ F_{337}\! \left(x \right) &= F_{338}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{338}\! \left(x \right) &= F_{339}\! \left(x \right)+F_{397}\! \left(x \right)\\ F_{339}\! \left(x \right) &= F_{340}\! \left(x \right)+F_{342}\! \left(x \right)\\ F_{340}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{341}\! \left(x \right)\\ F_{341}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{342}\! \left(x \right) &= F_{343}\! \left(x \right)+F_{358}\! \left(x \right)\\ F_{343}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{344}\! \left(x \right)+F_{357}\! \left(x \right)\\ F_{344}\! \left(x \right) &= F_{345}\! \left(x \right)\\ F_{345}\! \left(x \right) &= F_{346}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{346}\! \left(x \right) &= F_{347}\! \left(x \right)+F_{348}\! \left(x \right)\\ F_{347}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{343}\! \left(x \right)\\ F_{348}\! \left(x \right) &= F_{349}\! \left(x \right)+F_{352}\! \left(x \right)\\ F_{349}\! \left(x \right) &= F_{12}\! \left(x \right) F_{350}\! \left(x \right)\\ F_{350}\! \left(x \right) &= F_{347}\! \left(x \right)+F_{351}\! \left(x \right)\\ F_{351}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{352}\! \left(x \right) &= F_{353}\! \left(x \right)+F_{355}\! \left(x \right)\\ F_{353}\! \left(x \right) &= F_{354}\! \left(x \right)\\ F_{354}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{355}\! \left(x \right) &= F_{356}\! \left(x \right)\\ F_{356}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{357}\! \left(x \right) &= F_{166}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{358}\! \left(x \right) &= F_{359}\! \left(x \right)+F_{389}\! \left(x \right)\\ F_{359}\! \left(x \right) &= F_{360}\! \left(x \right)\\ F_{360}\! \left(x \right) &= F_{361}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{361}\! \left(x \right) &= F_{362}\! \left(x \right)+F_{365}\! \left(x \right)\\ F_{362}\! \left(x \right) &= F_{363}\! \left(x \right)+F_{364}\! \left(x \right)\\ F_{363}\! \left(x \right) &= F_{359}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{364}\! \left(x \right) &= F_{12}\! \left(x \right) F_{213}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{365}\! \left(x \right) &= F_{12}\! \left(x \right) F_{366}\! \left(x \right)\\ F_{366}\! \left(x \right) &= F_{367}\! \left(x \right)+F_{388}\! \left(x \right)\\ F_{367}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{368}\! \left(x \right)\\ F_{368}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{369}\! \left(x \right)+F_{385}\! \left(x \right)\\ F_{369}\! \left(x \right) &= F_{370}\! \left(x \right)\\ F_{370}\! \left(x \right) &= F_{371}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{371}\! \left(x \right) &= F_{367}\! \left(x \right)+F_{372}\! \left(x \right)\\ F_{372}\! \left(x \right) &= F_{12}\! \left(x \right) F_{373}\! \left(x \right)\\ F_{373}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{374}\! \left(x \right)\\ F_{374}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{375}\! \left(x \right)\\ F_{375}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{376}\! \left(x \right)\\ F_{376}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{377}\! \left(x \right)+F_{379}\! \left(x \right)+F_{380}\! \left(x \right)\\ F_{377}\! \left(x \right) &= F_{378}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{378}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{376}\! \left(x \right)\\ F_{379}\! \left(x \right) &= F_{375}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{380}\! \left(x \right) &= F_{381}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{381}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{382}\! \left(x \right)\\ F_{382}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{380}\! \left(x \right)+F_{383}\! \left(x \right)\\ F_{383}\! \left(x \right) &= F_{384}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{384}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{382}\! \left(x \right)\\ F_{385}\! \left(x \right) &= F_{386}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{386}\! \left(x \right) &= F_{387}\! \left(x \right)\\ F_{387}\! \left(x \right) &= F_{11}\! \left(x \right) F_{211}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{388}\! \left(x \right) &= F_{12}\! \left(x \right) F_{215}\! \left(x \right)\\ F_{389}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{390}\! \left(x \right)+F_{392}\! \left(x \right)\\ F_{390}\! \left(x \right) &= F_{391}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{391}\! \left(x \right) &= F_{389}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{392}\! \left(x \right) &= F_{393}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{393}\! \left(x \right) &= F_{394}\! \left(x \right)+F_{395}\! \left(x \right)\\ F_{394}\! \left(x \right) &= F_{12}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{395}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{396}\! \left(x \right)\\ F_{396}\! \left(x \right) &= F_{170}\! \left(x \right)+F_{389}\! \left(x \right)\\ F_{397}\! \left(x \right) &= F_{12}\! \left(x \right) F_{398}\! \left(x \right)\\ F_{398}\! \left(x \right) &= F_{399}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{399}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{400}\! \left(x \right)+F_{417}\! \left(x \right)+F_{424}\! \left(x \right)\\ F_{400}\! \left(x \right) &= F_{4}\! \left(x \right) F_{401}\! \left(x \right)\\ F_{401}\! \left(x \right) &= F_{402}\! \left(x \right)+F_{403}\! \left(x \right)\\ F_{402}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{399}\! \left(x \right)\\ F_{403}\! \left(x \right) &= F_{404}\! \left(x \right)+F_{407}\! \left(x \right)\\ F_{404}\! \left(x \right) &= F_{405}\! \left(x \right)\\ F_{405}\! \left(x \right) &= F_{4}\! \left(x \right) F_{406}\! \left(x \right)\\ F_{406}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{404}\! \left(x \right)\\ F_{407}\! \left(x \right) &= F_{408}\! \left(x \right)\\ F_{408}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{409}\! \left(x \right)\\ F_{409}\! \left(x \right) &= F_{410}\! \left(x \right)+F_{414}\! \left(x \right)\\ F_{410}\! \left(x \right) &= F_{411}\! \left(x \right)+F_{412}\! \left(x \right)\\ F_{411}\! \left(x \right) &= F_{12}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{412}\! \left(x \right) &= F_{11}\! \left(x \right) F_{413}\! \left(x \right)\\ F_{413}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{414}\! \left(x \right) &= F_{415}\! \left(x \right)+F_{416}\! \left(x \right)\\ F_{415}\! \left(x \right) &= F_{25}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{416}\! \left(x \right) &= F_{24}\! \left(x \right) F_{413}\! \left(x \right)\\ F_{417}\! \left(x \right) &= F_{4}\! \left(x \right) F_{418}\! \left(x \right)\\ F_{418}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{419}\! \left(x \right)\\ F_{419}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{325}\! \left(x \right)+F_{420}\! \left(x \right)+F_{422}\! \left(x \right)\\ F_{420}\! \left(x \right) &= F_{4}\! \left(x \right) F_{421}\! \left(x \right)\\ F_{421}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{419}\! \left(x \right)\\ F_{422}\! \left(x \right) &= F_{4}\! \left(x \right) F_{423}\! \left(x \right)\\ F_{423}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{419}\! \left(x \right)\\ F_{424}\! \left(x \right) &= F_{398}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements Req Corrob" and has 356 rules.

Found on January 17, 2022.

Finding the specification took 234 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{10}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{26}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{10}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{61}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{10}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= 2 F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\ F_{89}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{9}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{9}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{99}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{100}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{10} \left(x \right)^{3}\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{10}\! \left(x \right) F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{109}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{10}\! \left(x \right) F_{115}\! \left(x \right) F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{10}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{128}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{12}\! \left(x \right) F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{127}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{12}\! \left(x \right) F_{125}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{126}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{12}\! \left(x \right) F_{125}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{12}\! \left(x \right) F_{122}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{12}\! \left(x \right) F_{130}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{131}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{12}\! \left(x \right) F_{125}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{176}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{169}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{12}\! \left(x \right) F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{10}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{12}\! \left(x \right) F_{147}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{12}\! \left(x \right) F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{10}\! \left(x \right) F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{17}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{64}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{10}\! \left(x \right) F_{163}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{164}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{165}\! \left(x \right)+F_{167}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{12}\! \left(x \right) F_{166}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{12}\! \left(x \right) F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{10}\! \left(x \right) F_{170}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{12}\! \left(x \right) F_{172}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{174}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{170}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{10}\! \left(x \right) F_{175}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{308}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{12}\! \left(x \right) F_{180}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)+F_{191}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{188}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{12}\! \left(x \right) F_{183}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{184}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{12}\! \left(x \right) F_{186}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{184}\! \left(x \right)+F_{187}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{12}\! \left(x \right) F_{186}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{12}\! \left(x \right) F_{190}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{188}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)+F_{193}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{145}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{223}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{195}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{12}\! \left(x \right) F_{196}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{204}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{10}\! \left(x \right) F_{199}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{203}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{19}\! \left(x \right)+F_{201}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{12}\! \left(x \right) F_{200}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{10}\! \left(x \right) F_{125}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)+F_{208}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{206}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{207}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{26}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{212}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{10}\! \left(x \right) F_{210}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{211}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{213}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{214}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{10}\! \left(x \right) F_{215}\! \left(x \right)\\ F_{215}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{216}\! \left(x \right)+F_{221}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{12}\! \left(x \right) F_{217}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{215}\! \left(x \right)+F_{218}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{12}\! \left(x \right) F_{220}\! \left(x \right)\\ F_{220}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{218}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{12}\! \left(x \right) F_{222}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{215}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{223}\! \left(x \right) &= F_{224}\! \left(x \right)\\ F_{224}\! \left(x \right) &= F_{12}\! \left(x \right) F_{225}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{226}\! \left(x \right)+F_{229}\! \left(x \right)\\ F_{226}\! \left(x \right) &= F_{145}\! \left(x \right) F_{227}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{228}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{21}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{230}\! \left(x \right)+F_{242}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{231}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{232}\! \left(x \right)+F_{233}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{234}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{235}\! \left(x \right)+F_{240}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{12}\! \left(x \right) F_{236}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{237}\! \left(x \right)+F_{239}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{238}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{12}\! \left(x \right) F_{236}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{12}\! \left(x \right) F_{186}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{241}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{12}\! \left(x \right) F_{231}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{243}\! \left(x \right)+F_{252}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{244}\! \left(x \right)+F_{248}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{246}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{10}\! \left(x \right) F_{145}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{247}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{10} \left(x \right)^{3} F_{9}\! \left(x \right)\\ F_{248}\! \left(x \right) &= F_{249}\! \left(x \right)+F_{250}\! \left(x \right)\\ F_{249}\! \left(x \right) &= F_{10}\! \left(x \right) F_{150}\! \left(x \right)\\ F_{250}\! \left(x \right) &= F_{251}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{259}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{254}\! \left(x \right)+F_{255}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{10}\! \left(x \right) F_{194}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{256}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{257}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{258}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{260}\! \left(x \right)+F_{305}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{10}\! \left(x \right) F_{261}\! \left(x \right)\\ F_{261}\! \left(x \right) &= F_{262}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{12}\! \left(x \right) F_{263}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{264}\! \left(x \right)+F_{304}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{265}\! \left(x \right)\\ F_{265}\! \left(x \right) &= F_{10}\! \left(x \right) F_{266}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{267}\! \left(x \right)+F_{295}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{268}\! \left(x \right)+F_{270}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{269}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{156}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{271}\! \left(x \right)+F_{272}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{10}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{272}\! \left(x \right) &= F_{273}\! \left(x \right)+F_{287}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{10}\! \left(x \right) F_{274}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{275}\! \left(x \right)+F_{286}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{276}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{12}\! \left(x \right) F_{277}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{278}\! \left(x \right)+F_{285}\! \left(x \right)\\ F_{278}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{279}\! \left(x \right)+F_{280}\! \left(x \right)\\ F_{279}\! \left(x \right) &= F_{12}\! \left(x \right) F_{200}\! \left(x \right)\\ F_{280}\! \left(x \right) &= F_{281}\! \left(x \right)\\ F_{281}\! \left(x \right) &= F_{12}\! \left(x \right) F_{282}\! \left(x \right)\\ F_{282}\! \left(x \right) &= F_{283}\! \left(x \right)+F_{284}\! \left(x \right)\\ F_{283}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{284}\! \left(x \right) &= F_{278}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{285}\! \left(x \right) &= F_{12}\! \left(x \right) F_{200}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{286}\! \left(x \right) &= F_{12}\! \left(x \right) F_{120}\! \left(x \right)\\ F_{287}\! \left(x \right) &= F_{288}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{288}\! \left(x \right) &= F_{289}\! \left(x \right)\\ F_{289}\! \left(x \right) &= F_{12}\! \left(x \right) F_{290}\! \left(x \right)\\ F_{290}\! \left(x \right) &= F_{291}\! \left(x \right)+F_{294}\! \left(x \right)\\ F_{291}\! \left(x \right) &= F_{18}\! \left(x \right) F_{292}\! \left(x \right)\\ F_{292}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{293}\! \left(x \right)\\ F_{293}\! \left(x \right) &= F_{10}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{294}\! \left(x \right) &= F_{21}\! \left(x \right) F_{282}\! \left(x \right)\\ F_{295}\! \left(x \right) &= F_{10}\! \left(x \right) F_{296}\! \left(x \right)\\ F_{296}\! \left(x \right) &= F_{297}\! \left(x \right)+F_{299}\! \left(x \right)\\ F_{297}\! \left(x \right) &= F_{298}\! \left(x \right)\\ F_{298}\! \left(x \right) &= F_{156}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{299}\! \left(x \right) &= F_{300}\! \left(x \right)+F_{301}\! \left(x \right)\\ F_{300}\! \left(x \right) &= F_{9}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{301}\! \left(x \right) &= F_{272}\! \left(x \right)+F_{302}\! \left(x \right)\\ F_{302}\! \left(x \right) &= F_{303}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{303}\! \left(x \right) &= F_{164}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{304}\! \left(x \right) &= F_{10}\! \left(x \right) F_{204}\! \left(x \right)\\ F_{305}\! \left(x \right) &= F_{306}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{306}\! \left(x \right) &= F_{307}\! \left(x \right)\\ F_{307}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{227}\! \left(x \right) F_{258}\! \left(x \right)\\ F_{308}\! \left(x \right) &= F_{309}\! \left(x \right)+F_{321}\! \left(x \right)\\ F_{309}\! \left(x \right) &= F_{310}\! \left(x \right)\\ F_{310}\! \left(x \right) &= F_{12}\! \left(x \right) F_{311}\! \left(x \right)\\ F_{311}\! \left(x \right) &= F_{312}\! \left(x \right)+F_{318}\! \left(x \right)\\ F_{312}\! \left(x \right) &= F_{309}\! \left(x \right)+F_{313}\! \left(x \right)\\ F_{313}\! \left(x \right) &= F_{314}\! \left(x \right)\\ F_{314}\! \left(x \right) &= F_{12}\! \left(x \right) F_{315}\! \left(x \right)\\ F_{315}\! \left(x \right) &= F_{316}\! \left(x \right)+F_{317}\! \left(x \right)\\ F_{316}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{317}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{312}\! \left(x \right)\\ F_{318}\! \left(x \right) &= F_{319}\! \left(x \right)+F_{320}\! \left(x \right)\\ F_{319}\! \left(x \right) &= F_{26}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{320}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{261}\! \left(x \right)\\ F_{321}\! \left(x \right) &= F_{322}\! \left(x \right)\\ F_{322}\! \left(x \right) &= F_{12}\! \left(x \right) F_{323}\! \left(x \right)\\ F_{323}\! \left(x \right) &= F_{324}\! \left(x \right)+F_{340}\! \left(x \right)\\ F_{324}\! \left(x \right) &= F_{325}\! \left(x \right)+F_{326}\! \left(x \right)\\ F_{325}\! \left(x \right) &= F_{156}\! \left(x \right) F_{170}\! \left(x \right)\\ F_{326}\! \left(x \right) &= F_{321}\! \left(x \right)+F_{327}\! \left(x \right)\\ F_{327}\! \left(x \right) &= F_{10}\! \left(x \right) F_{328}\! \left(x \right)\\ F_{328}\! \left(x \right) &= F_{321}\! \left(x \right)+F_{329}\! \left(x \right)\\ F_{329}\! \left(x \right) &= F_{330}\! \left(x \right)\\ F_{330}\! \left(x \right) &= F_{12}\! \left(x \right) F_{331}\! \left(x \right)\\ F_{331}\! \left(x \right) &= F_{332}\! \left(x \right)+F_{333}\! \left(x \right)\\ F_{332}\! \left(x \right) &= F_{170}\! \left(x \right)+F_{329}\! \left(x \right)\\ F_{333}\! \left(x \right) &= F_{334}\! \left(x \right)+F_{335}\! \left(x \right)\\ F_{334}\! \left(x \right) &= F_{170}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{335}\! \left(x \right) &= F_{336}\! \left(x \right)+F_{338}\! \left(x \right)\\ F_{336}\! \left(x \right) &= F_{337}\! \left(x \right)\\ F_{337}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{9}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{338}\! \left(x \right) &= F_{339}\! \left(x \right)\\ F_{339}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right) F_{9}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{340}\! \left(x \right) &= F_{341}\! \left(x \right)+F_{345}\! \left(x \right)\\ F_{341}\! \left(x \right) &= F_{170}\! \left(x \right) F_{342}\! \left(x \right)\\ F_{342}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{343}\! \left(x \right)\\ F_{343}\! \left(x \right) &= F_{10}\! \left(x \right) F_{344}\! \left(x \right)\\ F_{344}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{345}\! \left(x \right) &= F_{346}\! \left(x \right)+F_{347}\! \left(x \right)\\ F_{346}\! \left(x \right) &= F_{156}\! \left(x \right) F_{336}\! \left(x \right)\\ F_{347}\! \left(x \right) &= F_{348}\! \left(x \right)+F_{354}\! \left(x \right)\\ F_{348}\! \left(x \right) &= F_{349}\! \left(x \right)\\ F_{349}\! \left(x \right) &= F_{12}\! \left(x \right) F_{350}\! \left(x \right) F_{9}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{350}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{240}\! \left(x \right)+F_{351}\! \left(x \right)\\ F_{351}\! \left(x \right) &= F_{12}\! \left(x \right) F_{352}\! \left(x \right)\\ F_{352}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{237}\! \left(x \right)+F_{353}\! \left(x \right)\\ F_{353}\! \left(x \right) &= F_{12}\! \left(x \right) F_{183}\! \left(x \right)\\ F_{354}\! \left(x \right) &= F_{10}\! \left(x \right) F_{355}\! \left(x \right)\\ F_{355}\! \left(x \right) &= F_{338}\! \left(x \right)+F_{348}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements" and has 370 rules.

Found on January 17, 2022.

Finding the specification took 170 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{34}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{10}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{53}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{12}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{12}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{10}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{34}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{34}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{35}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{10}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{24}\! \left(x \right)+F_{97}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{12}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{106}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{101}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{153}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{10}\! \left(x \right) F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{123}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{122}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{12}\! \left(x \right) F_{120}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{121}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{12}\! \left(x \right) F_{120}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{117}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{12}\! \left(x \right) F_{125}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{126}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{12}\! \left(x \right) F_{120}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{10}\! \left(x \right) F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{12}\! \left(x \right) F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{138}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{22}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{144}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{12}\! \left(x \right) F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{142}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{12}\! \left(x \right) F_{141}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{12}\! \left(x \right) F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{139}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{10}\! \left(x \right) F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{10}\! \left(x \right) F_{141}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{120}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{10}\! \left(x \right) F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{163}\! \left(x \right)\\ F_{158}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{159}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{12}\! \left(x \right) F_{160}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{12}\! \left(x \right) F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{12}\! \left(x \right) F_{165}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{170}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{168}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{12}\! \left(x \right) F_{169}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{142}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{169}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{227}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{178}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\ F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{219}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{180}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{12}\! \left(x \right) F_{181}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{186}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)+F_{184}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{176}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{185}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{188}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{204}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{12}\! \left(x \right) F_{191}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{192}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{193}\! \left(x \right)+F_{197}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{195}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{196}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{10}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{201}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{12}\! \left(x \right) F_{200}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{198}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{203}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{12}\! \left(x \right) F_{206}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{207}\! \left(x \right)+F_{218}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{10}\! \left(x \right) F_{208}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{211}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{210}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{212}\! \left(x \right)+F_{215}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{213}\! \left(x \right)+F_{214}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\ F_{214}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{216}\! \left(x \right)+F_{217}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{158}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{10}\! \left(x \right) F_{193}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{10}\! \left(x \right) F_{220}\! \left(x \right)\\ F_{220}\! \left(x \right) &= F_{221}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{12}\! \left(x \right) F_{222}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{224}\! \left(x \right)\\ F_{223}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{220}\! \left(x \right)\\ F_{224}\! \left(x \right) &= F_{10}\! \left(x \right) F_{225}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{226}\! \left(x \right)\\ F_{226}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{228}\! \left(x \right)+F_{345}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{229}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{230}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{12}\! \left(x \right) F_{231}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{232}\! \left(x \right)+F_{242}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{233}\! \left(x \right)+F_{239}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{12}\! \left(x \right) F_{234}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{233}\! \left(x \right)+F_{235}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{236}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{12}\! \left(x \right) F_{237}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{238}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{12}\! \left(x \right) F_{237}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{240}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{12}\! \left(x \right) F_{241}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{238}\! \left(x \right)+F_{239}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{243}\! \left(x \right)+F_{244}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{189}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{266}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{246}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{12}\! \left(x \right) F_{247}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{248}\! \left(x \right)+F_{252}\! \left(x \right)\\ F_{248}\! \left(x \right) &= F_{249}\! \left(x \right)\\ F_{249}\! \left(x \right) &= F_{10}\! \left(x \right) F_{250}\! \left(x \right)\\ F_{250}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{251}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{10}\! \left(x \right) F_{120}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{256}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{254}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{255}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{36}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{257}\! \left(x \right)+F_{258}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{10}\! \left(x \right) F_{195}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{259}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{260}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{10}\! \left(x \right) F_{261}\! \left(x \right)\\ F_{261}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{262}\! \left(x \right)+F_{264}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{12}\! \left(x \right) F_{263}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{261}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{12}\! \left(x \right) F_{265}\! \left(x \right)\\ F_{265}\! \left(x \right) &= F_{261}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{267}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{12}\! \left(x \right) F_{268}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{269}\! \left(x \right)+F_{272}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{189}\! \left(x \right) F_{270}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{271}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{26}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{272}\! \left(x \right) &= F_{273}\! \left(x \right)+F_{285}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{274}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{275}\! \left(x \right)+F_{276}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{10}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{277}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{278}\! \left(x \right)+F_{283}\! \left(x \right)\\ F_{278}\! \left(x \right) &= F_{12}\! \left(x \right) F_{279}\! \left(x \right)\\ F_{279}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{280}\! \left(x \right)+F_{282}\! \left(x \right)\\ F_{280}\! \left(x \right) &= F_{281}\! \left(x \right)\\ F_{281}\! \left(x \right) &= F_{12}\! \left(x \right) F_{279}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{282}\! \left(x \right) &= F_{12}\! \left(x \right) F_{237}\! \left(x \right)\\ F_{283}\! \left(x \right) &= F_{284}\! \left(x \right)\\ F_{284}\! \left(x \right) &= F_{12}\! \left(x \right) F_{274}\! \left(x \right)\\ F_{285}\! \left(x \right) &= F_{286}\! \left(x \right)+F_{295}\! \left(x \right)\\ F_{286}\! \left(x \right) &= F_{287}\! \left(x \right)+F_{291}\! \left(x \right)\\ F_{287}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{289}\! \left(x \right)\\ F_{288}\! \left(x \right) &= F_{10}\! \left(x \right) F_{189}\! \left(x \right)\\ F_{289}\! \left(x \right) &= F_{290}\! \left(x \right)\\ F_{290}\! \left(x \right) &= F_{10} \left(x \right)^{3} F_{9}\! \left(x \right)\\ F_{291}\! \left(x \right) &= F_{292}\! \left(x \right)+F_{293}\! \left(x \right)\\ F_{292}\! \left(x \right) &= F_{10}\! \left(x \right) F_{204}\! \left(x \right)\\ F_{293}\! \left(x \right) &= F_{294}\! \left(x \right)\\ F_{294}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{295}\! \left(x \right) &= F_{296}\! \left(x \right)+F_{302}\! \left(x \right)\\ F_{296}\! \left(x \right) &= F_{297}\! \left(x \right)+F_{298}\! \left(x \right)\\ F_{297}\! \left(x \right) &= F_{10}\! \left(x \right) F_{245}\! \left(x \right)\\ F_{298}\! \left(x \right) &= F_{299}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{299}\! \left(x \right) &= F_{300}\! \left(x \right)\\ F_{300}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{301}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{301}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{302}\! \left(x \right) &= F_{303}\! \left(x \right)+F_{333}\! \left(x \right)\\ F_{303}\! \left(x \right) &= F_{10}\! \left(x \right) F_{304}\! \left(x \right)\\ F_{304}\! \left(x \right) &= F_{305}\! \left(x \right)\\ F_{305}\! \left(x \right) &= F_{12}\! \left(x \right) F_{306}\! \left(x \right)\\ F_{306}\! \left(x \right) &= F_{307}\! \left(x \right)+F_{332}\! \left(x \right)\\ F_{307}\! \left(x \right) &= F_{308}\! \left(x \right)\\ F_{308}\! \left(x \right) &= F_{10}\! \left(x \right) F_{309}\! \left(x \right)\\ F_{309}\! \left(x \right) &= F_{310}\! \left(x \right)+F_{325}\! \left(x \right)\\ F_{310}\! \left(x \right) &= F_{311}\! \left(x \right)+F_{313}\! \left(x \right)\\ F_{311}\! \left(x \right) &= F_{312}\! \left(x \right)\\ F_{312}\! \left(x \right) &= F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{313}\! \left(x \right) &= F_{314}\! \left(x \right)+F_{315}\! \left(x \right)\\ F_{314}\! \left(x \right) &= F_{10}\! \left(x \right) F_{212}\! \left(x \right)\\ F_{315}\! \left(x \right) &= F_{316}\! \left(x \right)+F_{317}\! \left(x \right)\\ F_{316}\! \left(x \right) &= F_{10}\! \left(x \right) F_{167}\! \left(x \right)\\ F_{317}\! \left(x \right) &= F_{318}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{318}\! \left(x \right) &= F_{319}\! \left(x \right)\\ F_{319}\! \left(x \right) &= F_{12}\! \left(x \right) F_{320}\! \left(x \right)\\ F_{320}\! \left(x \right) &= F_{321}\! \left(x \right)+F_{324}\! \left(x \right)\\ F_{321}\! \left(x \right) &= F_{23}\! \left(x \right) F_{322}\! \left(x \right)\\ F_{322}\! \left(x \right) &= F_{323}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{323}\! \left(x \right) &= F_{10}\! \left(x \right) F_{226}\! \left(x \right)\\ F_{324}\! \left(x \right) &= F_{146}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{325}\! \left(x \right) &= F_{10}\! \left(x \right) F_{326}\! \left(x \right)\\ F_{326}\! \left(x \right) &= F_{327}\! \left(x \right)+F_{329}\! \left(x \right)\\ F_{327}\! \left(x \right) &= F_{328}\! \left(x \right)\\ F_{328}\! \left(x \right) &= F_{18}\! \left(x \right) F_{226}\! \left(x \right)\\ F_{329}\! \left(x \right) &= F_{330}\! \left(x \right)+F_{331}\! \left(x \right)\\ F_{330}\! \left(x \right) &= F_{212}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{331}\! \left(x \right) &= F_{215}\! \left(x \right)+F_{315}\! \left(x \right)\\ F_{332}\! \left(x \right) &= F_{10}\! \left(x \right) F_{252}\! \left(x \right)\\ F_{333}\! \left(x \right) &= F_{334}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{334}\! \left(x \right) &= F_{335}\! \left(x \right)\\ F_{335}\! \left(x \right) &= F_{12}\! \left(x \right) F_{336}\! \left(x \right)\\ F_{336}\! \left(x \right) &= F_{337}\! \left(x \right)+F_{338}\! \left(x \right)\\ F_{337}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right) F_{270}\! \left(x \right)\\ F_{338}\! \left(x \right) &= F_{339}\! \left(x \right)+F_{341}\! \left(x \right)\\ F_{339}\! \left(x \right) &= F_{340}\! \left(x \right)\\ F_{340}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{341}\! \left(x \right) &= F_{333}\! \left(x \right)+F_{342}\! \left(x \right)\\ F_{342}\! \left(x \right) &= F_{10}\! \left(x \right) F_{343}\! \left(x \right)\\ F_{343}\! \left(x \right) &= F_{344}\! \left(x \right)\\ F_{344}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{345}\! \left(x \right) &= F_{346}\! \left(x \right)+F_{359}\! \left(x \right)\\ F_{346}\! \left(x \right) &= F_{347}\! \left(x \right)\\ F_{347}\! \left(x \right) &= F_{12}\! \left(x \right) F_{348}\! \left(x \right)\\ F_{348}\! \left(x \right) &= F_{349}\! \left(x \right)+F_{356}\! \left(x \right)\\ F_{349}\! \left(x \right) &= F_{346}\! \left(x \right)+F_{350}\! \left(x \right)\\ F_{350}\! \left(x \right) &= F_{351}\! \left(x \right)\\ F_{351}\! \left(x \right) &= F_{12}\! \left(x \right) F_{352}\! \left(x \right)\\ F_{352}\! \left(x \right) &= F_{353}\! \left(x \right)+F_{354}\! \left(x \right)\\ F_{353}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{354}\! \left(x \right) &= F_{349}\! \left(x \right)+F_{355}\! \left(x \right)\\ F_{355}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{356}\! \left(x \right) &= F_{357}\! \left(x \right)+F_{358}\! \left(x \right)\\ F_{357}\! \left(x \right) &= F_{226}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{358}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{304}\! \left(x \right)\\ F_{359}\! \left(x \right) &= F_{360}\! \left(x \right)\\ F_{360}\! \left(x \right) &= F_{12}\! \left(x \right) F_{361}\! \left(x \right)\\ F_{361}\! \left(x \right) &= F_{362}\! \left(x \right)+F_{367}\! \left(x \right)\\ F_{362}\! \left(x \right) &= F_{359}\! \left(x \right)+F_{363}\! \left(x \right)\\ F_{363}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{283}\! \left(x \right)+F_{364}\! \left(x \right)\\ F_{364}\! \left(x \right) &= F_{12}\! \left(x \right) F_{365}\! \left(x \right)\\ F_{365}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{280}\! \left(x \right)+F_{366}\! \left(x \right)\\ F_{366}\! \left(x \right) &= F_{12}\! \left(x \right) F_{234}\! \left(x \right)\\ F_{367}\! \left(x \right) &= F_{368}\! \left(x \right)+F_{369}\! \left(x \right)\\ F_{368}\! \left(x \right) &= F_{226}\! \left(x \right) F_{363}\! \left(x \right)\\ F_{369}\! \left(x \right) &= F_{299}\! \left(x \right)+F_{334}\! \left(x \right)\\ \end{align*}\)

Av(1234, 3412)

This specification was found using the strategy pack "Insertion Point Col Placements" and has 261 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Insertion Point Row Placements" and has 278 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Insertion Row And Col Placements" and has 425 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Insertion Point Placements Req Corrob" and has 356 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Insertion Point Placements" and has 370 rules.

Proof Tree

System of Equations