Av(1243, 4231)
Generating Function
$$\displaystyle \frac{4 x^{10}-127 x^{9}+692 x^{8}-1657 x^{7}+2305 x^{6}-2045 x^{5}+1196 x^{4}-461 x^{3}+113 x^{2}-16 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{6}}$$
Counting Sequence
1, 1, 2, 6, 22, 86, 335, 1266, 4598, 16016, 53579, 172663, 537957, 1626504, 4789128, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{6} F \! \left(x \right)-4 x^{10}+127 x^{9}-692 x^{8}+1657 x^{7}-2305 x^{6}+2045 x^{5}-1196 x^{4}+461 x^{3}-113 x^{2}+16 x -1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 22$$
$$\displaystyle a \! \left(5\right) = 86$$
$$\displaystyle a \! \left(6\right) = 335$$
$$\displaystyle a \! \left(7\right) = 1266$$
$$\displaystyle a \! \left(8\right) = 4598$$
$$\displaystyle a \! \left(9\right) = 16016$$
$$\displaystyle a \! \left(10\right) = 53579$$
$$\displaystyle a \! \left(n +8\right) = -64 a \! \left(n \right)+384 a \! \left(n +1\right)-880 a \! \left(n +2\right)+1072 a \! \left(n +3\right)-780 a \! \left(n +4\right)+352 a \! \left(n +5\right)-97 a \! \left(n +6\right)+15 a \! \left(n +7\right)+5 n -5, \quad n \geq 11$$
Explicit Closed Form
$$\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(4224 \sqrt{5}+9600\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{3840}+\frac{\left(-4224 \sqrt{5}+9600\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{3840}+\\\frac{\left(n^{5}+30 n^{4}-805 n^{3}+6870 n^{2}-42336 n +61200\right) 2^{n}}{3840}-5 n -20 & \text{otherwise} \end{array}\right.$$
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 Row And Col Placements Req Corrob" and has 278 rules.

Found on April 26, 2021.

Finding the specification took 87 seconds.

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Copy 278 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{34}\! \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_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \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_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{51}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{23}\! \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_{71}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{67}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{11}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{43}\! \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_{41}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{96}\! \left(x \right) &= 3 F_{23}\! \left(x \right)+F_{97}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{103}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \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_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{116}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{12}\! \left(x \right)\\ 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_{18}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{123}\! \left(x \right)+F_{23}\! \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_{19}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{131}\! \left(x \right)+F_{133}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{128}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{126}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{146}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{139}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{113}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{150}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{12}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{154}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{113}\! \left(x \right) F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right) F_{22}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{163}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{12}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{165}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{19}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{59}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{168}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{171}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{170}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{119}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{15}\! \left(x \right) F_{174}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{197}\! \left(x \right)+F_{199}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{178}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{178}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{175}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{180}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{181}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{188}\! \left(x \right)\\ F_{182}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{183}\! \left(x \right)+F_{186}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{184}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{183}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{188}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{189}\! \left(x \right)+F_{194}\! \left(x \right)+F_{196}\! \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_{67}\! \left(x \right)\\ F_{191}\! \left(x \right) &= 3 F_{23}\! \left(x \right)+F_{189}\! \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_{185}\! \left(x \right)+F_{191}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{195}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{188}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{198}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{174}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{200}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{201}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)+F_{205}\! \left(x \right)\\ F_{202}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{183}\! \left(x \right)+F_{203}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{202}\! \left(x \right)\\ F_{205}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{189}\! \left(x \right)+F_{206}\! \left(x \right)+F_{208}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{207}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{205}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{201}\! \left(x \right) F_{4}\! \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_{35}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{213}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{214}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{215}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{216}\! \left(x \right)+F_{233}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{217}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{12}\! \left(x \right) F_{218}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{220}\! \left(x \right)\\ F_{220}\! \left(x \right) &= F_{221}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{218}\! \left(x \right)+F_{222}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{219}\! \left(x \right)+F_{223}\! \left(x \right)\\ F_{223}\! \left(x \right) &= F_{224}\! \left(x \right)+F_{227}\! \left(x \right)\\ F_{224}\! \left(x \right) &= F_{225}\! \left(x \right)+F_{226}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{19}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{226}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{19}\! \left(x \right) F_{228}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{229}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{230}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{231}\! \left(x \right) F_{232}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{234}\! \left(x \right)+F_{277}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{237}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{236}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{113}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{238}\! \left(x \right)+F_{241}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{23}\! \left(x \right)+F_{239}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{240}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{238}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{242}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{243}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{244}\! \left(x \right)+F_{245}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{229}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{246}\! \left(x \right)+F_{273}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{247}\! \left(x \right)+F_{272}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{15}\! \left(x \right) F_{248}\! \left(x \right)\\ F_{248}\! \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_{252}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{248}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{248}\! \left(x \right)+F_{253}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{254}\! \left(x \right)+F_{257}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{255}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{256}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{39}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{258}\! \left(x \right)+F_{266}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{259}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{260}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{261}\! \left(x \right)+F_{262}\! \left(x \right)\\ F_{261}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{263}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{264}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{265}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{265}\! \left(x \right) &= F_{263}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{267}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{19}\! \left(x \right) F_{268}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{269}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{27}\! \left(x \right) F_{270}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{271}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{272}\! \left(x \right) &= F_{18}\! \left(x \right) F_{268}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{274}\! \left(x \right)+F_{275}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{15}\! \left(x \right) F_{241}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{276}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right) F_{268}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{213}\! \left(x \right)\\ \end{align*}

This specification was found using the strategy pack "Insertion Row And Col Placements" and has 279 rules.

Found on April 26, 2021.

Finding the specification took 86 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_{35}\! \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_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{53}\! \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_{25}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{49}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{210}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{67}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{77}\! \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_{13}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{41}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{89}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{97}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{100}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{104}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{117}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{124}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{121}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{132}\! \left(x \right)+F_{134}\! \left(x \right)+F_{21}\! \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_{20}\! \left(x \right)\\ F_{129}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{127}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{140}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{36}\! \left(x \right) F_{57}\! \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_{143}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{114}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{17} \left(x \right)^{2} F_{26}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{151}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{57}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \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_{114}\! \left(x \right) F_{14}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{26}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{167}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{57}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{17}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{59}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{174}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{120}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{13}\! \left(x \right) F_{175}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{198}\! \left(x \right)+F_{200}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{179}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{176}\! \left(x \right)+F_{180}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)+F_{189}\! \left(x \right)\\ F_{183}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{184}\! \left(x \right)+F_{187}\! \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_{64}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{184}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{188}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{183}\! \left(x \right)\\ F_{189}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{190}\! \left(x \right)+F_{195}\! \left(x \right)+F_{197}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{191}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{192}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{190}\! \left(x \right)+F_{193}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{186}\! \left(x \right)+F_{192}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{196}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{189}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{175}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{201}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{202}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{203}\! \left(x \right)+F_{206}\! \left(x \right)\\ F_{203}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{184}\! \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_{120}\! \left(x \right)+F_{203}\! \left(x \right)\\ F_{206}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{190}\! \left(x \right)+F_{207}\! \left(x \right)+F_{209}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{208}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{206}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{202}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{211}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{212}\! \left(x \right)+F_{213}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{214}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{215}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{216}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{217}\! \left(x \right)+F_{234}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{218}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{219}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{220}\! \left(x \right)+F_{36}\! \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_{219}\! \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_{225}\! \left(x \right)+F_{228}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{226}\! \left(x \right)+F_{227}\! \left(x \right)\\ F_{226}\! \left(x \right) &= F_{17}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{17}\! \left(x \right) F_{229}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{230}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{231}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{232}\! \left(x \right) F_{233}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{278}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{236}\! \left(x \right)+F_{238}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{237}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{114}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{239}\! \left(x \right)+F_{242}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{21}\! \left(x \right)+F_{240}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{241}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{239}\! \left(x \right)\\ F_{242}\! \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_{246}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{16}\! \left(x \right) F_{230}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{247}\! \left(x \right)+F_{274}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{248}\! \left(x \right)+F_{273}\! \left(x \right)\\ F_{248}\! \left(x \right) &= F_{13}\! \left(x \right) F_{249}\! \left(x \right)\\ F_{249}\! \left(x \right) &= F_{250}\! \left(x \right)\\ F_{250}\! \left(x \right) &= F_{251}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{252}\! \left(x \right)+F_{253}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{249}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{249}\! \left(x \right)+F_{254}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{255}\! \left(x \right)+F_{258}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{256}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{257}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{17} \left(x \right)^{2} F_{26}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{259}\! \left(x \right)+F_{267}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{260}\! \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_{40}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{264}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{265}\! \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_{40}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{268}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{17}\! \left(x \right) F_{269}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{270}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{271}\! \left(x \right) F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{272}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{272}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{16}\! \left(x \right) F_{269}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{275}\! \left(x \right)+F_{276}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{13}\! \left(x \right) F_{242}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{277}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{16}\! \left(x \right) F_{269}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{278}\! \left(x \right) &= F_{214}\! \left(x \right)\\ \end{align*}