Av(1234, 2431)
View Raw Data
Generating Function
\(\displaystyle -\frac{11 x^{9}-65 x^{8}+195 x^{7}-365 x^{6}+447 x^{5}-359 x^{4}+189 x^{3}-63 x^{2}+12 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 22, 86, 330, 1198, 4087, 13185, 40619, 120636, 348197, 983073, 2728722, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{6} F \! \left(x \right)+11 x^{9}-65 x^{8}+195 x^{7}-365 x^{6}+447 x^{5}-359 x^{4}+189 x^{3}-63 x^{2}+12 x -1 = 0\)
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 86\)
\(\displaystyle a \! \left(6\right) = 330\)
\(\displaystyle a \! \left(7\right) = 1198\)
\(\displaystyle a \! \left(8\right) = 4087\)
\(\displaystyle a \! \left(9\right) = 13185\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{5}}{120}-\frac{n^{4}}{24}-\frac{5 n^{3}}{24}+\frac{73 n^{2}}{24}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{73 n}{60}+2, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \frac{\left(12 \sqrt{5}+540\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(-12 \sqrt{5}+540\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{5}}{120}+\frac{n^{4}}{12}+\frac{23 n^{3}}{24}+\frac{23 n^{2}}{12}-3 \,2^{-1+n} n +\frac{391 n}{30}-18 \,2^{n}+10\)

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 "Point Placements" and has 268 rules.

Found on January 18, 2022.

Finding the specification took 29 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_{37}\! \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_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \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_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{265}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{47}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{53}\! \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_{19}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{234}\! \left(x \right)+F_{262}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{15}\! \left(x \right)+F_{163}\! \left(x \right)+F_{231}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{72}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{72}\! \left(x \right) &= 0\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{77}\! \left(x \right) &= 0\\ F_{78}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{115}\! \left(x \right)+F_{15}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{84}\! \left(x \right) &= 0\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{89}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{104}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{115}\! \left(x \right) &= 0\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{120}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{116}\! \left(x \right)+F_{125}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{131}\! \left(x \right)+F_{135}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{135}\! \left(x \right) &= 0\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{144}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{152}\! \left(x \right)+F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{147}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{151}\! \left(x \right)\\ F_{156}\! \left(x \right) &= 0\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{165}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{166}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{167}\! \left(x \right)+F_{213}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{168}\! \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_{14}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{108}\! \left(x \right)+F_{172}\! \left(x \right)+F_{194}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{178}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{179}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{180}\! \left(x \right)+F_{184}\! \left(x \right)+F_{185}\! \left(x \right)+F_{193}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{183}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{175}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{179}\! \left(x \right)\\ F_{184}\! \left(x \right) &= 0\\ F_{185}\! \left(x \right) &= F_{186}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{188}\! \left(x \right)\\ F_{188}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{189}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{190}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{191}\! \left(x \right)+F_{192}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{188}\! \left(x \right)\\ F_{193}\! \left(x \right) &= 0\\ F_{194}\! \left(x \right) &= F_{195}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{196}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{197}\! \left(x \right)\\ F_{197}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{136}\! \left(x \right)+F_{198}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{201}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{202}\! \left(x \right)\\ F_{202}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{194}\! \left(x \right)+F_{203}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)+F_{207}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{206}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{176}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{202}\! \left(x \right)+F_{208}\! \left(x \right)\\ F_{208}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{135}\! \left(x \right)+F_{184}\! \left(x \right)+F_{185}\! \left(x \right)+F_{209}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{210}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{212}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{206}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{208}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{214}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{215}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{216}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{217}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{218}\! \left(x \right)+F_{219}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{215}\! \left(x \right)+F_{220}\! \left(x \right)\\ F_{220}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{160}\! \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_{225}\! \left(x \right)\\ F_{223}\! \left(x \right) &= F_{224}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{224}\! \left(x \right) &= F_{148}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{220}\! \left(x \right)+F_{226}\! \left(x \right)\\ F_{226}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{156}\! \left(x \right)+F_{157}\! \left(x \right)+F_{227}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{228}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{229}\! \left(x \right)+F_{230}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{224}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{226}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{232}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{233}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{235}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{236}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{237}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{238}\! \left(x \right)+F_{260}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{239}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{240}\! \left(x \right)+F_{241}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{237}\! \left(x \right)+F_{242}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{213}\! \left(x \right)+F_{234}\! \left(x \right)+F_{243}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{244}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{247}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{246}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{242}\! \left(x \right)+F_{248}\! \left(x \right)\\ F_{248}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{116}\! \left(x \right)+F_{15}\! \left(x \right)+F_{163}\! \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_{251}\! \left(x \right)+F_{254}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{246}\! \left(x \right)+F_{252}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{253}\! \left(x \right)+F_{72}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{253}\! \left(x \right) &= 0\\ F_{254}\! \left(x \right) &= F_{248}\! \left(x \right)+F_{255}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{15}\! \left(x \right)+F_{193}\! \left(x \right)+F_{256}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{257}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{258}\! \left(x \right)+F_{259}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{252}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{255}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{261}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{261}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{263}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{233}\! \left(x \right)+F_{264}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{215}\! \left(x \right)\\ F_{265}\! \left(x \right) &= F_{266}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{264}\! \left(x \right)+F_{267}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{237}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 393 rules.

Found on April 28, 2021.

Finding the specification took 13 seconds.

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