###### Av(1234, 1342)
Counting Sequence
1, 1, 2, 6, 22, 89, 380, 1678, 7584, 34875, 162560, 766124, 3644066, 17469863, 84324840, ...
Implicit Equation for the Generating Function
$$\displaystyle x \left(x^{2}-2 x +2\right) F \left(x \right)^{4}+\left(2 x^{2}-4 x -1\right) F \left(x \right)^{3}+\left(2 x +3\right) F \left(x \right)^{2}-3 F \! \left(x \right)+1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 22$$
$$\displaystyle a \! \left(5\right) = 89$$
$$\displaystyle a \! \left(6\right) = 380$$
$$\displaystyle a \! \left(7\right) = 1678$$
$$\displaystyle a \! \left(n +8\right) = -\frac{2 \left(4 n +5\right) \left(2 n +3\right) \left(4 n +3\right) a \! \left(n \right)}{9 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(14276 n^{3}+105300 n^{2}+252862 n +196515\right) a \! \left(n +1\right)}{27 \left(n +9\right) \left(n +7\right) \left(n +6\right)}-\frac{\left(88628 n^{3}+756114 n^{2}+2152783 n +2046777\right) a \! \left(n +2\right)}{54 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(271825 n^{3}+2607174 n^{2}+8461187 n +9320358\right) a \! \left(n +3\right)}{108 \left(n +9\right) \left(n +7\right) \left(n +6\right)}-\frac{\left(200183 n^{3}+2331978 n^{2}+9033313 n +11721078\right) a \! \left(n +4\right)}{108 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(20005 n^{3}+283212 n^{2}+1323137 n +2050536\right) a \! \left(n +5\right)}{27 \left(n +9\right) \left(n +7\right) \left(n +6\right)}-\frac{2 \left(2258 n^{3}+37968 n^{2}+210241 n +384615\right) a \! \left(n +6\right)}{27 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(181 n^{2}+2436 n +7931\right) a \! \left(n +7\right)}{9 \left(n +7\right) \left(n +9\right)}, \quad n \geq 8$$
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 "Point Placements Tracked Fusion Req Corrob Expand Verified" and has 223 rules.

Found on January 27, 2022.

Finding the specification took 15092 seconds.

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Copy 223 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_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= -\frac{y \left(F_{14}\! \left(x , 1\right)-F_{14}\! \left(x , y\right)\right)}{-1+y}\\ F_{23}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{220}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x , 1\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{215}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{195}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= -\frac{-y F_{35}\! \left(x , y\right)+F_{35}\! \left(x , 1\right)}{-1+y}\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{40}\! \left(x , y\right) &= F_{189}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x \right) F_{52}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x , 1\right)\\ F_{45}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{43}\! \left(x \right)\\ F_{50}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= -\frac{y \left(F_{45}\! \left(x , 1\right)-F_{45}\! \left(x , y\right)\right)}{-1+y}\\ F_{52}\! \left(x , y\right) &= F_{185}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{161}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)^{2} F_{17}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)^{2} F_{17}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{59}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{59}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{63}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{64}\! \left(x , y\right) &= F_{160}\! \left(x \right) F_{63}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{159}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{72}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{70}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{77}\! \left(x , y\right) &= -\frac{-y F_{78}\! \left(x , y\right)+F_{78}\! \left(x , 1\right)}{-1+y}\\ F_{78}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{149}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{83}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x \right)+F_{97}\! \left(x , y\right)\\ F_{84}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{73}\! \left(x , 1\right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{44}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{9}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x , 1\right)\\ F_{95}\! \left(x , y\right) &= F_{96}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{45}\! \left(x , y\right) F_{63}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{97}\! \left(x , y\right) &= F_{98}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{101}\! \left(x , y\right)\\ F_{100}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{84}\! \left(x \right)\\ F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)\\ F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{104}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{129}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)\\ F_{106}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)+F_{128}\! \left(x , y\right)\\ F_{107}\! \left(x , y\right) &= F_{106}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)\\ F_{109}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\ F_{110}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{112}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= F_{110}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{112}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{113}\! \left(x , y\right)\\ F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{115}\! \left(x , y\right) &= F_{116}\! \left(x \right)+F_{117}\! \left(x , y\right)+F_{120}\! \left(x , y\right)\\ F_{116}\! \left(x \right) &= 0\\ F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{117}\! \left(x , y\right)\\ F_{120}\! \left(x , y\right) &= F_{114}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)+F_{124}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{123}\! \left(x , y\right)\\ F_{123}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{2}\! \left(x \right)\\ F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)\\ F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{127}\! \left(x , y\right) &= F_{126}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{127}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{128}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{84}\! \left(x \right)\\ F_{129}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\ F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)+F_{132}\! \left(x , y\right)\\ F_{131}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{92}\! \left(x \right)\\ F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)\\ F_{133}\! \left(x , y\right) &= F_{134}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{134}\! \left(x , y\right) &= F_{135}\! \left(x , y\right)+F_{139}\! \left(x , y\right)\\ F_{135}\! \left(x , y\right) &= F_{136}\! \left(x , y\right) F_{92}\! \left(x \right)\\ F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\ F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)\\ F_{138}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\ F_{139}\! \left(x , y\right) &= F_{140}\! \left(x , y\right) F_{91}\! \left(x \right)\\ F_{140}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)\\ F_{141}\! \left(x , y\right) &= F_{142}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{142}\! \left(x , y\right) &= F_{143}\! \left(x , y\right)\\ F_{143}\! \left(x , y\right) &= F_{144}\! \left(x , y\right)+F_{146}\! \left(x , y\right)\\ F_{144}\! \left(x , y\right) &= F_{145}\! \left(x \right) F_{18}\! \left(x , y\right)\\ F_{145}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ F_{146}\! \left(x , y\right) &= F_{147}\! \left(x , y\right)\\ F_{147}\! \left(x , y\right) &= F_{148}\! \left(x \right) F_{63}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{149}\! \left(x , y\right) &= F_{150}\! \left(x , y\right)\\ F_{150}\! \left(x , y\right) &= F_{151}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{151}\! \left(x , y\right) &= F_{152}\! \left(x , y\right)+F_{157}\! \left(x , y\right)\\ F_{152}\! \left(x , y\right) &= F_{153}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{153}\! \left(x , y\right) &= F_{154}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{154}\! \left(x , y\right) &= F_{155}\! \left(x , y\right)\\ F_{155}\! \left(x , y\right) &= F_{156}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{156}\! \left(x , y\right) &= -\frac{y \left(F_{153}\! \left(x , 1\right)-F_{153}\! \left(x , y\right)\right)}{-1+y}\\ F_{157}\! \left(x , y\right) &= F_{158}\! \left(x , y\right)\\ F_{158}\! \left(x , y\right) &= F_{153}\! \left(x , y\right) F_{63}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{159}\! \left(x , y\right) &= F_{160}\! \left(x \right) F_{18}\! \left(x , y\right)\\ F_{160}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ F_{161}\! \left(x , y\right) &= F_{162}\! \left(x , y\right)+F_{172}\! \left(x , y\right)\\ F_{162}\! \left(x , y\right) &= F_{163}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{163}\! \left(x , y\right) &= F_{164}\! \left(x , y\right)\\ F_{164}\! \left(x , y\right) &= F_{165}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{165}\! \left(x , y\right) &= F_{166}\! \left(x , y\right)+F_{170}\! \left(x , y\right)\\ F_{166}\! \left(x , y\right) &= F_{167}\! \left(x , y\right) F_{84}\! \left(x \right)\\ F_{167}\! \left(x , y\right) &= F_{168}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{168}\! \left(x , y\right) &= F_{169}\! \left(x , y\right)\\ F_{169}\! \left(x , y\right) &= F_{53}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{170}\! \left(x , y\right) &= F_{171}\! \left(x , y\right)\\ F_{171}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{167}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{172}\! \left(x , y\right) &= F_{173}\! \left(x , y\right)\\ F_{173}\! \left(x , y\right) &= F_{174}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{175}\! \left(x , y\right) &= F_{174}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{175}\! \left(x , y\right) &= F_{176}\! \left(x , y\right)\\ F_{177}\! \left(x , y\right) &= F_{163}\! \left(x , y\right)+F_{176}\! \left(x , y\right)\\ F_{177}\! \left(x , y\right) &= F_{178}\! \left(x , y\right)\\ F_{178}\! \left(x , y\right) &= F_{179}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{179}\! \left(x , y\right) &= F_{180}\! \left(x , y\right)+F_{183}\! \left(x , y\right)\\ F_{180}\! \left(x , y\right) &= F_{181}\! \left(x , y\right) F_{84}\! \left(x \right)\\ F_{182}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{181}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{182}\! \left(x , y\right) &= F_{168}\! \left(x , y\right)\\ F_{183}\! \left(x , y\right) &= F_{184}\! \left(x , y\right)\\ F_{184}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{181}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{186}\! \left(x , y\right) &= F_{185}\! \left(x , y\right)+F_{188}\! \left(x , y\right)\\ F_{187}\! \left(x , y\right) &= F_{186}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{187}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\ F_{188}\! \left(x , y\right) &= F_{37}\! \left(x \right) F_{52}\! \left(x , y\right)\\ F_{190}\! \left(x , y\right) &= F_{189}\! \left(x , y\right)+F_{193}\! \left(x , y\right)\\ F_{191}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{190}\! \left(x , y\right)\\ F_{191}\! \left(x , y\right) &= F_{192}\! \left(x , y\right)\\ F_{181}\! \left(x , y\right) &= F_{192}\! \left(x , y\right)+F_{43}\! \left(x \right)\\ F_{193}\! \left(x , y\right) &= F_{194}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{194}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{43}\! \left(x \right)\\ F_{196}\! \left(x , y\right) &= F_{195}\! \left(x , y\right)+F_{205}\! \left(x , y\right)\\ F_{197}\! \left(x , y\right) &= F_{196}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{197}\! \left(x , y\right) &= F_{198}\! \left(x , y\right)\\ F_{198}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{199}\! \left(x , y\right)\\ F_{199}\! \left(x , y\right) &= F_{200}\! \left(x , y\right)+F_{202}\! \left(x , y\right)\\ F_{200}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{201}\! \left(x , y\right)\\ F_{201}\! \left(x , y\right) &= F_{37}\! \left(x \right)+F_{46}\! \left(x , y\right)\\ F_{202}\! \left(x , y\right) &= F_{203}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{203}\! \left(x , y\right) &= F_{204}\! \left(x , y\right)\\ F_{204}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{63}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{205}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{206}\! \left(x , y\right)\\ F_{207}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{206}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{207}\! \left(x , y\right) &= F_{208}\! \left(x , y\right)\\ F_{208}\! \left(x , y\right) &= F_{201}\! \left(x , y\right)+F_{209}\! \left(x , y\right)\\ F_{209}\! \left(x , y\right) &= F_{210}\! \left(x , y\right)\\ F_{210}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{211}\! \left(x , y\right)\\ F_{211}\! \left(x , y\right) &= F_{200}\! \left(x , y\right)+F_{212}\! \left(x , y\right)\\ F_{212}\! \left(x , y\right) &= F_{213}\! \left(x , y\right)\\ F_{213}\! \left(x , y\right) &= F_{214}\! \left(x , y\right) F_{52}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{214}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{215}\! \left(x , y\right) &= F_{216}\! \left(x , y\right)\\ F_{216}\! \left(x , y\right) &= F_{217}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{218}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{217}\! \left(x , y\right)\\ F_{218}\! \left(x , y\right) &= F_{219}\! \left(x , y\right)\\ F_{219}\! \left(x , y\right) &= -\frac{y \left(F_{29}\! \left(x , 1\right)-F_{29}\! \left(x , y\right)\right)}{-1+y}\\ F_{220}\! \left(x \right) &= F_{221}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{222}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{190}\! \left(x , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Insertion Row And Col Placements Tracked Fusion Req Corrob" and has 144 rules.

Found on April 24, 2021.

Finding the specification took 51369 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_{19}\! \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_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= -\frac{y \left(F_{27}\! \left(x , 1\right)-F_{27}\! \left(x , y\right)\right)}{-1+y}\\ F_{27}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= y x\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{140}\! \left(x , y\right)+F_{24}\! \left(x \right)+F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{49}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{11}\! \left(x \right)+F_{52}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{58}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{7}\! \left(x \right)\\ F_{68}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , 1, y\right)\\ F_{71}\! \left(x , y , z\right) &= F_{72}\! \left(x , y , z\right)\\ F_{72}\! \left(x , y , z\right) &= F_{138}\! \left(x , z , y\right)+F_{24}\! \left(x \right)+F_{73}\! \left(x , y , z\right)+F_{96}\! \left(x , y , z\right)\\ F_{73}\! \left(x , y , z\right) &= -\frac{y \left(F_{74}\! \left(x , 1, z\right)-F_{74}\! \left(x , y , z\right)\right)}{-1+y}\\ F_{74}\! \left(x , y , z\right) &= F_{4}\! \left(x \right) F_{75}\! \left(x , y , z\right)\\ F_{75}\! \left(x , y , z\right) &= F_{72}\! \left(x , y , z\right)+F_{76}\! \left(x , y , z\right)\\ F_{76}\! \left(x , y , z\right) &= F_{24}\! \left(x \right)+F_{77}\! \left(x , y , z\right)+F_{94}\! \left(x , z , y\right)\\ F_{77}\! \left(x , y , z\right) &= F_{30}\! \left(x , y\right) F_{78}\! \left(x , y , z\right)\\ F_{78}\! \left(x , y , z\right) &= F_{79}\! \left(x , y , z\right)+F_{83}\! \left(x , y , z\right)\\ F_{79}\! \left(x , y , z\right) &= F_{33}\! \left(x , z\right)+F_{80}\! \left(x , y , z\right)\\ F_{80}\! \left(x , y , z\right) &= F_{81}\! \left(x , z , y\right)\\ F_{81}\! \left(x , y , z\right) &= F_{30}\! \left(x , y\right) F_{82}\! \left(x , z , y\right)\\ F_{82}\! \left(x , y , z\right) &= F_{33}\! \left(x , y\right)+F_{80}\! \left(x , y , z\right)\\ F_{83}\! \left(x , y , z\right) &= F_{84}\! \left(x , y , z\right)+F_{89}\! \left(x , y , z\right)\\ F_{84}\! \left(x , y , z\right) &= F_{85}\! \left(x , y , z\right)\\ F_{85}\! \left(x , y , z\right) &= F_{30}\! \left(x , y\right) F_{86}\! \left(x , y , z\right)\\ F_{86}\! \left(x , y , z\right) &= F_{87}\! \left(x , y , z\right)+F_{88}\! \left(x , y , z\right)\\ F_{87}\! \left(x , y , z\right) &= F_{33}\! \left(x , z\right)\\ F_{88}\! \left(x , y , z\right) &= F_{84}\! \left(x , y , z\right)\\ F_{89}\! \left(x , y , z\right) &= F_{90}\! \left(x , y , z\right)\\ F_{90}\! \left(x , y , z\right) &= F_{30}\! \left(x , y\right) F_{91}\! \left(x , y , z\right)\\ F_{91}\! \left(x , y , z\right) &= F_{92}\! \left(x , y , z\right)+F_{93}\! \left(x , y , z\right)\\ F_{92}\! \left(x , y , z\right) &= F_{80}\! \left(x , y , z\right)\\ F_{93}\! \left(x , y , z\right) &= F_{89}\! \left(x , y , z\right)\\ F_{94}\! \left(x , y , z\right) &= F_{30}\! \left(x , y\right) F_{95}\! \left(x , z , y\right)\\ F_{95}\! \left(x , y , z\right) &= F_{28}\! \left(x , y\right)+F_{76}\! \left(x , y , z\right)\\ F_{96}\! \left(x , y , z\right) &= \frac{z \left(F_{97}\! \left(x , y , 1\right)-F_{97}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\ F_{97}\! \left(x , y , z\right) &= F_{98}\! \left(x , y , y z \right)\\ F_{99}\! \left(x , y , z\right) &= F_{98}\! \left(x , y z , z\right)\\ F_{99}\! \left(x , y , z\right) &= F_{100}\! \left(x , y , z\right) F_{30}\! \left(x , z\right)\\ F_{100}\! \left(x , y , z\right) &= F_{101}\! \left(x , y , z\right)+F_{44}\! \left(x , z\right)\\ F_{101}\! \left(x , y , z\right) &= F_{102}\! \left(x , y , z\right)\\ F_{102}\! \left(x , y , z\right) &= F_{103}\! \left(x , y , z\right) F_{30}\! \left(x , z\right)\\ F_{103}\! \left(x , y , z\right) &= F_{100}\! \left(x , y , z\right)+F_{104}\! \left(x , y , z\right)\\ F_{104}\! \left(x , y , z\right) &= F_{105}\! \left(x , y z , z\right)\\ F_{106}\! \left(x , y , z\right) &= F_{105}\! \left(x , y , y z \right)\\ F_{106}\! \left(x , y , z\right) &= F_{107}\! \left(x , y , z\right)+F_{109}\! \left(x , y , z\right)\\ F_{107}\! \left(x , y , z\right) &= F_{108}\! \left(x , y , z\right)\\ F_{108}\! \left(x , y , z\right) &= F_{11}\! \left(x \right) F_{32}\! \left(x , y\right) F_{33}\! \left(x , z\right)\\ F_{109}\! \left(x , y , z\right) &= F_{110}\! \left(x , y , y z \right)\\ F_{111}\! \left(x , y , z\right) &= F_{110}\! \left(x , y z , z\right)\\ F_{111}\! \left(x , y , z\right) &= F_{112}\! \left(x , z\right)+F_{122}\! \left(x , y , z\right)\\ F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)\\ F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{118}\! \left(x , y\right)\\ F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , 1, y\right)\\ F_{100}\! \left(x , y , z\right) &= F_{117}\! \left(x , y z , z\right)\\ F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)\\ F_{120}\! \left(x , y\right) &= F_{11} \left(x \right)^{2} F_{33}\! \left(x , y\right)\\ F_{121}\! \left(x , y\right) &= F_{109}\! \left(x , 1, y\right)\\ F_{122}\! \left(x , y , z\right) &= F_{123}\! \left(x , y z , z\right)\\ F_{123}\! \left(x , y , z\right) &= F_{124}\! \left(x , y , z\right)\\ F_{124}\! \left(x , y , z\right) &= F_{125}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\ F_{125}\! \left(x , y , z\right) &= F_{126}\! \left(x , y , z\right)+F_{134}\! \left(x , y , z\right)\\ F_{126}\! \left(x , y , z\right) &= F_{127}\! \left(x , y , z\right)+F_{132}\! \left(x , y , z\right)\\ F_{127}\! \left(x , y , z\right) &= F_{128}\! \left(x , y , z\right)\\ F_{128}\! \left(x , y , z\right) &= F_{129}\! \left(x , y , z\right) F_{30}\! \left(x , y\right)\\ F_{129}\! \left(x , y , z\right) &= F_{130}\! \left(x , y , z\right)+F_{131}\! \left(x , y , z\right)\\ F_{130}\! \left(x , y , z\right) &= F_{56}\! \left(x , z\right)\\ F_{131}\! \left(x , y , z\right) &= F_{127}\! \left(x , y , z\right)\\ F_{132}\! \left(x , y , z\right) &= -\frac{y \left(F_{133}\! \left(x , 1, z\right)-F_{133}\! \left(x , y , z\right)\right)}{-1+y}\\ F_{101}\! \left(x , y , z\right) &= F_{133}\! \left(x , y z , z\right)\\ F_{134}\! \left(x , y , z\right) &= F_{135}\! \left(x , y , z\right)+F_{137}\! \left(x , y , z\right)\\ F_{135}\! \left(x , y , z\right) &= F_{136}\! \left(x , y , z\right)\\ F_{136}\! \left(x , y , z\right) &= F_{11} \left(x \right)^{2} F_{33}\! \left(x , y\right) F_{33}\! \left(x , z\right)\\ F_{137}\! \left(x , y , z\right) &= -\frac{y \left(F_{123}\! \left(x , 1, z\right)-F_{123}\! \left(x , y , z\right)\right)}{-1+y}\\ F_{138}\! \left(x , y , z\right) &= F_{139}\! \left(x , z , y\right) F_{30}\! \left(x , y\right)\\ F_{139}\! \left(x , y , z\right) &= F_{23}\! \left(x , y\right)+F_{72}\! \left(x , y , z\right)\\ F_{140}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\ F_{141}\! \left(x , y\right) &= F_{142}\! \left(x , 1, y\right)\\ F_{142}\! \left(x , y , z\right) &= F_{143}\! \left(x , y z , z\right)\\ F_{143}\! \left(x , y , z\right) &= F_{133}\! \left(x , y , z\right)+F_{23}\! \left(x , y\right)\\ \end{align*}

### This specification was found using the strategy pack "Row Placements Tracked Fusion Req Corrob" and has 42 rules.

Found on April 23, 2021.

Finding the specification took 94 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_{7}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{6}\! \left(x , y_{0}\right) F_{7}\! \left(x \right)\\ F_{6}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{4}\! \left(x , y_{0}\right)+F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x , y_{0}\right) &= F_{9}\! \left(x , y_{0}, 1\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{24}\! \left(x , y_{1}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{7}\! \left(x \right)\\ F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{17}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{7}\! \left(x \right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{17}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{24}\! \left(x , y_{0}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} F_{22}\! \left(x , y_{0}, y_{1}\right)-y_{2} F_{22}\! \left(x , y_{0}, y_{2}\right)}{-y_{2}+y_{1}}\\ F_{22}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{23}\! \left(x , y_{0}, 1\right)-y_{1} F_{23}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{24}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{1}\right) F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{26}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} F_{11}\! \left(x , 1, y_{1}, y_{2}\right)-y_{0} F_{11}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{2}\right) F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{30}\! \left(x , y_{0}, y_{2}\right)-y_{1} F_{30}\! \left(x , y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{32}\! \left(x , y_{0} y_{1}\right) y_{0}-y_{2} F_{32}\! \left(x , y_{1} y_{2}\right)}{y_{0}-y_{2}}\\ F_{32}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y_{0}\right)+F_{41}\! \left(x , y_{0}\right)\\ F_{33}\! \left(x , y_{0}\right) &= F_{34}\! \left(x , y_{0}\right) F_{7}\! \left(x \right)\\ F_{34}\! \left(x , y_{0}\right) &= F_{35}\! \left(x , 1, y_{0}\right)\\ F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y_{0}, y_{1}\right)+F_{38}\! \left(x , y_{0}, y_{1}\right)+F_{39}\! \left(x , y_{0}, y_{1}\right)\\ F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{37}\! \left(x , y_{0}, y_{1}\right) F_{7}\! \left(x \right)\\ F_{37}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{35}\! \left(x , y_{0}, y_{1}\right)+F_{35}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\ F_{38}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{1}\right) F_{24}\! \left(x , y_{0}\right)\\ F_{39}\! \left(x , y_{0}, y_{1}\right) &= F_{40}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{2}\right) F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{41}\! \left(x , y_{0}\right) &= F_{24}\! \left(x , y_{0}\right) F_{32}\! \left(x , y_{0}\right)\\ \end{align*}

### This specification was found using the strategy pack "Point Placements Tracked Fusion Expand Verified" and has 272 rules.

Found on January 27, 2022.

Finding the specification took 10112 seconds.

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Copy 272 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_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= y x\\ F_{17}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{269}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{255}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{214}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= -\frac{-y F_{32}\! \left(x , y\right)+F_{32}\! \left(x , 1\right)}{-1+y}\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{35}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{212}\! \left(x \right)+F_{36}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{206}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{46}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{46}\! \left(x , y\right) &= F_{123}\! \left(x , y\right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x , 1\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{58}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{61}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{68}\! \left(x , y\right) &= -\frac{-y F_{69}\! \left(x , y\right)+F_{69}\! \left(x , 1\right)}{-1+y}\\ F_{69}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\ F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{78}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{78}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{80}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\ F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{80}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\ F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{9}\! \left(x \right) F_{90}\! \left(x \right) F_{93}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{72}\! \left(x \right) F_{9}\! \left(x \right) F_{93}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{74}\! \left(x \right)\\ F_{96}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{98}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)+F_{103}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{102}\! \left(x \right) F_{41}\! \left(x , y\right)\\ F_{102}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{104}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{113}\! \left(x , y\right)\\ F_{105}\! \left(x , y\right) &= F_{104}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)\\ F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\ F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{110}\! \left(x , y\right)\\ F_{109}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{112}\! \left(x , y\right) &= -\frac{-y F_{108}\! \left(x , y\right)+F_{108}\! \left(x , 1\right)}{-1+y}\\ F_{113}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)\\ F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , y\right)\\ F_{117}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{9}\! \left(x \right) F_{93}\! \left(x , y\right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x , 1\right)\\ F_{121}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)+F_{122}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{126}\! \left(x , y\right) &= F_{125}\! \left(x , y\right) F_{135}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{126}\! \left(x , y\right) &= F_{127}\! \left(x , y\right)\\ F_{128}\! \left(x , y\right) &= F_{127}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\ F_{129}\! \left(x , y\right) &= F_{128}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)\\ F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)\\ F_{131}\! \left(x , y\right) &= F_{132}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{133}\! \left(x , y\right) &= F_{132}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{133}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{134}\! \left(x , y\right) &= F_{135}\! \left(x , y\right) F_{138}\! \left(x \right)\\ F_{136}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{135}\! \left(x , y\right)\\ F_{137}\! \left(x , y\right) &= F_{136}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{137}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\ F_{138}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{47}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{141}\! \left(x , y\right) &= F_{142}\! \left(x , y\right)\\ F_{142}\! \left(x , y\right) &= F_{143}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{144}\! \left(x , y\right) &= F_{143}\! \left(x , y\right) F_{41}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{144}\! \left(x , y\right) &= F_{145}\! \left(x , y\right)\\ F_{146}\! \left(x , y\right) &= F_{145}\! \left(x , y\right)+F_{205}\! \left(x , y\right)\\ F_{147}\! \left(x , y\right) &= F_{146}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{147}\! \left(x , y\right) &= F_{148}\! \left(x , y\right)\\ F_{148}\! \left(x , y\right) &= F_{149}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{149}\! \left(x , y\right) &= F_{150}\! \left(x , y\right)+F_{155}\! \left(x , y\right)\\ F_{150}\! \left(x , y\right) &= F_{151}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{151}\! \left(x , y\right) &= F_{152}\! \left(x , y\right)+F_{153}\! \left(x , y\right)\\ F_{152}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{153}\! \left(x , y\right) &= F_{154}\! \left(x , y\right)\\ F_{154}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right) F_{93}\! \left(x , y\right)\\ F_{155}\! \left(x , y\right) &= F_{156}\! \left(x , y\right)+F_{157}\! \left(x , y\right)\\ F_{156}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{158}\! \left(x , y\right) &= F_{157}\! \left(x , y\right)+F_{196}\! \left(x , y\right)\\ F_{158}\! \left(x , y\right) &= F_{159}\! \left(x , y\right)\\ F_{159}\! \left(x , y\right) &= F_{160}\! \left(x , y\right) F_{9}\! \left(x \right) F_{93}\! \left(x , y\right)\\ F_{161}\! \left(x , y\right) &= F_{160}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{162}\! \left(x , y\right) &= F_{158}\! \left(x , y\right)+F_{161}\! \left(x , y\right)\\ F_{163}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{162}\! \left(x , y\right)\\ F_{163}\! \left(x , y\right) &= F_{164}\! \left(x , y\right)\\ F_{165}\! \left(x , y\right) &= F_{164}\! \left(x , y\right)+F_{189}\! \left(x , y\right)\\ F_{166}\! \left(x , y\right) &= F_{165}\! \left(x , y\right)+F_{168}\! \left(x , y\right)\\ F_{167}\! \left(x , y\right) &= F_{166}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{167}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\ F_{168}\! \left(x , y\right) &= F_{169}\! \left(x , y\right)+F_{171}\! \left(x , y\right)\\ F_{169}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{170}\! \left(x , y\right)\\ F_{170}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{52}\! \left(x , y\right)\\ F_{171}\! \left(x , y\right) &= F_{172}\! \left(x , y\right)\\ F_{172}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{173}\! \left(x , y\right)\\ F_{173}\! \left(x , y\right) &= F_{174}\! \left(x , y\right)+F_{181}\! \left(x , y\right)\\ F_{174}\! \left(x , y\right) &= F_{175}\! \left(x , y\right)+F_{176}\! \left(x , y\right)\\ F_{175}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\ F_{176}\! \left(x , y\right) &= F_{177}\! \left(x , y\right)\\ F_{177}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)^{2} F_{16}\! \left(x , y\right) F_{178}\! \left(x , y\right)\\ F_{179}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{178}\! \left(x , y\right)\\ F_{179}\! \left(x , y\right) &= F_{180}\! \left(x , y\right)\\ F_{180}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{181}\! \left(x , y\right) &= F_{151}\! \left(x , y\right) F_{182}\! \left(x , y\right)\\ F_{182}\! \left(x , y\right) &= F_{183}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{183}\! \left(x , y\right) &= F_{184}\! \left(x \right)+F_{185}\! \left(x , y\right)+F_{188}\! \left(x , y\right)\\ F_{184}\! \left(x \right) &= 0\\ F_{185}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{186}\! \left(x , y\right)\\ F_{186}\! \left(x , y\right) &= F_{187}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{187}\! \left(x , y\right) &= F_{185}\! \left(x , y\right)\\ F_{188}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{182}\! \left(x , y\right)\\ F_{189}\! \left(x , y\right) &= F_{190}\! \left(x , y\right)\\ F_{190}\! \left(x , y\right) &= F_{191}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{191}\! \left(x , y\right) &= F_{192}\! \left(x , y\right)+F_{193}\! \left(x , y\right)\\ F_{192}\! \left(x , y\right) &= F_{170}\! \left(x , y\right) F_{47}\! \left(x \right)\\ F_{193}\! \left(x , y\right) &= F_{194}\! \left(x , y\right)\\ F_{194}\! \left(x , y\right) &= F_{125}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{195}\! \left(x , y\right)\\ F_{195}\! \left(x , y\right) &= F_{182}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{197}\! \left(x , y\right) &= F_{196}\! \left(x , y\right)+F_{204}\! \left(x , y\right)\\ F_{197}\! \left(x , y\right) &= F_{198}\! \left(x , y\right)\\ F_{198}\! \left(x , y\right) &= F_{199}\! \left(x , y\right) F_{9}\! \left(x \right) F_{93}\! \left(x , y\right)\\ F_{200}\! \left(x , y\right) &= F_{199}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{201}\! \left(x , y\right) &= F_{197}\! \left(x , y\right)+F_{200}\! \left(x , y\right)\\ F_{202}\! \left(x , y\right) &= F_{201}\! \left(x , y\right)+F_{203}\! \left(x , y\right)\\ F_{202}\! \left(x , y\right) &= F_{162}\! \left(x , y\right)+F_{173}\! \left(x , y\right)\\ F_{203}\! \left(x , y\right) &= F_{155}\! \left(x , y\right)+F_{174}\! \left(x , y\right)\\ F_{204}\! \left(x , y\right) &= F_{153}\! \left(x , y\right) F_{182}\! \left(x , y\right)\\ F_{205}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{206}\! \left(x , y\right) &= F_{207}\! \left(x , y\right)\\ F_{207}\! \left(x , y\right) &= F_{208}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{208}\! \left(x , y\right) &= F_{209}\! \left(x , y\right)+F_{210}\! \left(x , y\right)\\ F_{209}\! \left(x , y\right) &= F_{138}\! \left(x \right) F_{37}\! \left(x , y\right)\\ F_{210}\! \left(x , y\right) &= F_{211}\! \left(x , y\right)\\ F_{211}\! \left(x , y\right) &= F_{125}\! \left(x , y\right) F_{37}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{213}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{106}\! \left(x , 1\right)\\ F_{214}\! \left(x , y\right) &= F_{215}\! \left(x , y\right)\\ F_{215}\! \left(x , y\right) &= F_{216}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{216}\! \left(x , y\right) &= -\frac{-y F_{217}\! \left(x , y\right)+F_{217}\! \left(x , 1\right)}{-1+y}\\ F_{217}\! \left(x , y\right) &= F_{218}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{218}\! \left(x , y\right) &= F_{219}\! \left(x , y\right)\\ F_{219}\! \left(x , y\right) &= F_{220}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{221}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{220}\! \left(x , y\right)\\ F_{221}\! \left(x , y\right) &= F_{222}\! \left(x , y\right)\\ F_{222}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{223}\! \left(x , y\right)\\ F_{224}\! \left(x , y\right) &= F_{223}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{224}\! \left(x , y\right) &= F_{225}\! \left(x , y\right)\\ F_{226}\! \left(x , y\right) &= F_{225}\! \left(x , y\right)+F_{228}\! \left(x , y\right)\\ F_{227}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{226}\! \left(x , y\right)\\ F_{227}\! \left(x , y\right) &= F_{80}\! \left(x , y\right)\\ F_{229}\! \left(x , y\right) &= F_{228}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{229}\! \left(x , y\right) &= F_{230}\! \left(x , y\right)\\ F_{230}\! \left(x , y\right) &= F_{231}\! \left(x , y\right)+F_{72}\! \left(x \right)\\ F_{231}\! \left(x , y\right) &= F_{232}\! \left(x , y\right)\\ F_{232}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{233}\! \left(x , y\right)\\ F_{233}\! \left(x , y\right) &= F_{234}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\ F_{234}\! \left(x , y\right) &= F_{235}\! \left(x , y\right)\\ F_{235}\! \left(x , y\right) &= F_{236}\! \left(x , y\right) F_{72}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{236}\! \left(x , y\right) &= F_{237}\! \left(x , y\right)+F_{244}\! \left(x , y\right)\\ F_{237}\! \left(x , y\right) &= F_{238}\! \left(x , y\right)+F_{242}\! \left(x , y\right)\\ F_{238}\! \left(x , y\right) &= F_{230}\! \left(x , y\right)+F_{239}\! \left(x , y\right)\\ F_{239}\! \left(x , y\right) &= F_{240}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{240}\! \left(x , y\right) &= F_{241}\! \left(x , y\right)\\ F_{241}\! \left(x , y\right) &= F_{237}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{242}\! \left(x , y\right) &= F_{243}\! \left(x , y\right)\\ F_{243}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{236}\! \left(x , y\right)\\ F_{244}\! \left(x , y\right) &= F_{245}\! \left(x , y\right)\\ F_{245}\! \left(x , y\right) &= F_{246}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{247}\! \left(x , y\right) &= F_{246}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{247}\! \left(x , y\right) &= F_{248}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{248}\! \left(x , y\right)+F_{249}\! \left(x , y\right)\\ F_{249}\! \left(x , y\right) &= F_{250}\! \left(x , y\right)+F_{251}\! \left(x , y\right)\\ F_{250}\! \left(x , y\right) &= F_{239}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\ F_{251}\! \left(x , y\right) &= F_{252}\! \left(x , y\right)+F_{253}\! \left(x , y\right)\\ F_{252}\! \left(x , y\right) &= F_{213}\! \left(x \right) F_{41}\! \left(x , y\right)\\ F_{253}\! \left(x , y\right) &= F_{254}\! \left(x , y\right)\\ F_{254}\! \left(x , y\right) &= F_{213}\! \left(x \right) F_{236}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{255}\! \left(x , y\right) &= F_{256}\! \left(x , y\right)\\ F_{256}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{257}\! \left(x , y\right)\\ F_{258}\! \left(x , y\right) &= F_{257}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{258}\! \left(x , y\right) &= F_{259}\! \left(x , y\right)\\ F_{178}\! \left(x , y\right) &= F_{259}\! \left(x , y\right)+F_{260}\! \left(x , y\right)\\ F_{260}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{261}\! \left(x , y\right)\\ F_{261}\! \left(x , y\right) &= F_{262}\! \left(x , y\right)\\ F_{262}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{263}\! \left(x , y\right)\\ F_{264}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{263}\! \left(x , y\right)\\ F_{264}\! \left(x , y\right) &= F_{265}\! \left(x , y\right)\\ F_{265}\! \left(x , y\right) &= F_{266}\! \left(x , y\right)+F_{267}\! \left(x , y\right)\\ F_{266}\! \left(x , y\right) &= F_{106}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{267}\! \left(x , y\right) &= F_{268}\! \left(x , y\right)\\ F_{268}\! \left(x , y\right) &= F_{106}\! \left(x , y\right) F_{236}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{270}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{271}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{34}\! \left(x , 1\right)\\ \end{align*}