PermPal Database

Av(1234, 1432)

The requested set is not in the database, but a symmetry of it is.

Generating Function

\frac{x^{2} - 4 x + 1}{(4 x - 1) (x - 1)}

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Counting Sequence

1, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, ...

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Implicit Equation for the Generating Function

(4 x - 1) (x - 1) F (x) - x^{2} + 4 x - 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (n + 1) = 4 a (n) - 2, n \geq 3

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Explicit Closed Form

{\begin{array}{cc} 1 & n = 0 \\ \frac{2}{3} + \frac{4^{n}}{12} & otherwise \end{array}

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Heatmap

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

This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 160 rules.

Found on April 28, 2021.

Finding the specification took 4 seconds.

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\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) \\ F_{3} (x) & = F_{13} (x) F_{4} (x) \\ F_{4} (x) & = F_{0} (x) + F_{5} (x) \\ F_{5} (x) & = F_{32} (x) + F_{6} (x) \\ F_{6} (x) & = F_{7} (x) \\ F_{7} (x) & = F_{13} (x) F_{8} (x) \\ F_{8} (x) & = F_{10} (x) + F_{9} (x) \\ F_{9} (x) & = F_{1} (x) + F_{6} (x) \\ F_{10} (x) & = F_{11} (x) + F_{15} (x) \\ F_{11} (x) & = F_{12} (x) \\ F_{12} (x) & = F_{13} (x) F_{14} (x) \\ F_{13} (x) & = x \\ F_{14} (x) & = F_{1} (x) + F_{11} (x) \\ F_{15} (x) & = F_{16} (x) + F_{17} (x) + F_{19} (x) \\ F_{16} (x) & = 0 \\ F_{17} (x) & = F_{13} (x) F_{18} (x) \\ F_{18} (x) & = F_{15} (x) + F_{6} (x) \\ F_{19} (x) & = F_{13} (x) F_{20} (x) \\ F_{20} (x) & = F_{21} (x) + F_{25} (x) \\ F_{21} (x) & = F_{11} (x) + F_{22} (x) \\ F_{22} (x) & = F_{23} (x) \\ F_{23} (x) & = F_{13} (x) F_{24} (x) \\ F_{24} (x) & = F_{22} (x) + F_{6} (x) \\ F_{25} (x) & = F_{26} (x) + F_{29} (x) \\ F_{26} (x) & = F_{27} (x) \\ F_{27} (x) & = F_{13} (x) F_{28} (x) \\ F_{28} (x) & = F_{11} (x) + F_{26} (x) \\ F_{29} (x) & = F_{30} (x) \\ F_{30} (x) & = F_{13} (x) F_{31} (x) \\ F_{31} (x) & = F_{15} (x) + F_{29} (x) \\ F_{32} (x) & = F_{156} (x) + F_{16} (x) + F_{33} (x) \\ F_{33} (x) & = F_{13} (x) F_{34} (x) \\ F_{34} (x) & = F_{35} (x) + F_{36} (x) \\ F_{35} (x) & = F_{2} (x) + F_{32} (x) \\ F_{36} (x) & = F_{37} (x) + F_{63} (x) \\ F_{37} (x) & = F_{16} (x) + F_{38} (x) + F_{40} (x) \\ F_{38} (x) & = F_{13} (x) F_{39} (x) \\ F_{39} (x) & = F_{2} (x) + F_{37} (x) \\ F_{40} (x) & = F_{13} (x) F_{41} (x) \\ F_{41} (x) & = F_{42} (x) + F_{46} (x) \\ F_{42} (x) & = F_{11} (x) + F_{43} (x) \\ F_{43} (x) & = F_{44} (x) \\ F_{44} (x) & = F_{13} (x) F_{45} (x) \\ F_{45} (x) & = F_{2} (x) + F_{43} (x) \\ F_{46} (x) & = F_{15} (x) + F_{47} (x) \\ F_{47} (x) & = 2 F_{16} (x) + F_{48} (x) + F_{50} (x) \\ F_{48} (x) & = F_{13} (x) F_{49} (x) \\ F_{49} (x) & = F_{32} (x) + F_{47} (x) \\ F_{50} (x) & = F_{13} (x) F_{51} (x) \\ F_{51} (x) & = F_{52} (x) + F_{56} (x) \\ F_{52} (x) & = F_{43} (x) + F_{53} (x) \\ F_{53} (x) & = F_{54} (x) \\ F_{54} (x) & = F_{13} (x) F_{55} (x) \\ F_{55} (x) & = F_{32} (x) + F_{53} (x) \\ F_{56} (x) & = F_{57} (x) + F_{60} (x) \\ F_{57} (x) & = F_{58} (x) \\ F_{58} (x) & = F_{13} (x) F_{59} (x) \\ F_{59} (x) & = F_{37} (x) + F_{57} (x) \\ F_{60} (x) & = F_{61} (x) \\ F_{61} (x) & = F_{13} (x) F_{62} (x) \\ F_{62} (x) & = F_{60} (x) + F_{63} (x) \\ F_{63} (x) & = F_{16} (x) + F_{50} (x) + F_{64} (x) + F_{66} (x) \\ F_{64} (x) & = F_{13} (x) F_{65} (x) \\ F_{65} (x) & = F_{32} (x) + F_{63} (x) \\ F_{66} (x) & = F_{13} (x) F_{67} (x) \\ F_{67} (x) & = F_{68} (x) + F_{90} (x) \\ F_{68} (x) & = F_{15} (x) + F_{69} (x) \\ F_{69} (x) & = 2 F_{16} (x) + F_{70} (x) + F_{77} (x) \\ F_{70} (x) & = F_{13} (x) F_{71} (x) \\ F_{71} (x) & = F_{69} (x) + F_{72} (x) \\ F_{72} (x) & = F_{73} (x) \\ F_{73} (x) & = F_{13} (x) F_{74} (x) \\ F_{74} (x) & = F_{75} (x) + F_{76} (x) \\ F_{75} (x) & = F_{2} (x) + F_{72} (x) \\ F_{76} (x) & = F_{43} (x) + F_{69} (x) \\ F_{77} (x) & = F_{13} (x) F_{78} (x) \\ F_{78} (x) & = F_{79} (x) + F_{83} (x) \\ F_{79} (x) & = F_{43} (x) + F_{80} (x) \\ F_{80} (x) & = F_{81} (x) \\ F_{81} (x) & = F_{13} (x) F_{82} (x) \\ F_{82} (x) & = F_{72} (x) + F_{80} (x) \\ F_{83} (x) & = F_{84} (x) + F_{87} (x) \\ F_{84} (x) & = F_{85} (x) \\ F_{85} (x) & = F_{13} (x) F_{86} (x) \\ F_{86} (x) & = F_{43} (x) + F_{84} (x) \\ F_{87} (x) & = F_{88} (x) \\ F_{88} (x) & = F_{13} (x) F_{89} (x) \\ F_{89} (x) & = F_{69} (x) + F_{87} (x) \\ F_{90} (x) & = F_{122} (x) + F_{91} (x) \\ F_{91} (x) & = F_{109} (x) + F_{16} (x) + F_{92} (x) + F_{99} (x) \\ F_{92} (x) & = F_{13} (x) F_{93} (x) \\ F_{93} (x) & = F_{91} (x) + F_{94} (x) \\ F_{94} (x) & = F_{16} (x) + F_{19} (x) + F_{95} (x) \\ F_{95} (x) & = F_{13} (x) F_{96} (x) \\ F_{96} (x) & = F_{97} (x) + F_{98} (x) \\ F_{97} (x) & = F_{6} (x) + F_{94} (x) \\ F_{98} (x) & = F_{15} (x) + F_{91} (x) \\ F_{99} (x) & = F_{100} (x) F_{13} (x) \\ F_{100} (x) & = F_{101} (x) + F_{105} (x) \\ F_{101} (x) & = F_{102} (x) + F_{22} (x) \\ F_{102} (x) & = F_{103} (x) \\ F_{103} (x) & = F_{104} (x) F_{13} (x) \\ F_{104} (x) & = F_{102} (x) + F_{94} (x) \\ F_{105} (x) & = F_{106} (x) + F_{29} (x) \\ F_{106} (x) & = F_{107} (x) \\ F_{107} (x) & = F_{108} (x) F_{13} (x) \\ F_{108} (x) & = F_{106} (x) + F_{91} (x) \\ F_{109} (x) & = F_{110} (x) F_{13} (x) \\ F_{110} (x) & = F_{111} (x) + F_{115} (x) \\ F_{111} (x) & = F_{112} (x) + F_{26} (x) \\ F_{112} (x) & = F_{113} (x) \\ F_{113} (x) & = F_{114} (x) F_{13} (x) \\ F_{114} (x) & = F_{112} (x) + F_{22} (x) \\ F_{115} (x) & = F_{116} (x) + F_{119} (x) \\ F_{116} (x) & = F_{117} (x) \\ F_{117} (x) & = F_{118} (x) F_{13} (x) \\ F_{118} (x) & = F_{116} (x) + F_{26} (x) \\ F_{119} (x) & = F_{120} (x) \\ F_{120} (x) & = F_{121} (x) F_{13} (x) \\ F_{121} (x) & = F_{119} (x) + F_{29} (x) \\ F_{122} (x) & = 2 F_{16} (x) + F_{123} (x) + F_{130} (x) + F_{143} (x) \\ F_{123} (x) & = F_{124} (x) F_{13} (x) \\ F_{124} (x) & = F_{122} (x) + F_{125} (x) \\ F_{125} (x) & = 2 F_{16} (x) + F_{126} (x) + F_{50} (x) \\ F_{126} (x) & = F_{127} (x) F_{13} (x) \\ F_{127} (x) & = F_{128} (x) + F_{129} (x) \\ F_{128} (x) & = F_{125} (x) + F_{32} (x) \\ F_{129} (x) & = F_{122} (x) + F_{47} (x) \\ F_{130} (x) & = F_{13} (x) F_{131} (x) \\ F_{131} (x) & = F_{132} (x) + F_{136} (x) \\ F_{132} (x) & = F_{133} (x) + F_{53} (x) \\ F_{133} (x) & = F_{134} (x) \\ F_{134} (x) & = F_{13} (x) F_{135} (x) \\ F_{135} (x) & = F_{125} (x) + F_{133} (x) \\ F_{136} (x) & = F_{137} (x) + F_{140} (x) \\ F_{137} (x) & = F_{138} (x) \\ F_{138} (x) & = F_{13} (x) F_{139} (x) \\ F_{139} (x) & = F_{137} (x) + F_{47} (x) \\ F_{140} (x) & = F_{141} (x) \\ F_{141} (x) & = F_{13} (x) F_{142} (x) \\ F_{142} (x) & = F_{122} (x) + F_{140} (x) \\ F_{143} (x) & = F_{13} (x) F_{144} (x) \\ F_{144} (x) & = F_{145} (x) + F_{149} (x) \\ F_{145} (x) & = F_{146} (x) + F_{84} (x) \\ F_{146} (x) & = F_{147} (x) \\ F_{147} (x) & = F_{13} (x) F_{148} (x) \\ F_{148} (x) & = F_{146} (x) + F_{53} (x) \\ F_{149} (x) & = F_{150} (x) + F_{153} (x) \\ F_{150} (x) & = F_{151} (x) \\ F_{151} (x) & = F_{13} (x) F_{152} (x) \\ F_{152} (x) & = F_{150} (x) + F_{57} (x) \\ F_{153} (x) & = F_{154} (x) \\ F_{154} (x) & = F_{13} (x) F_{155} (x) \\ F_{155} (x) & = F_{153} (x) + F_{60} (x) \\ F_{156} (x) & = F_{13} (x) F_{157} (x) \\ F_{157} (x) & = F_{158} (x) + F_{159} (x) \\ F_{158} (x) & = F_{6} (x) + F_{72} (x) \\ F_{159} (x) & = F_{125} (x) + F_{94} (x) \end{aligned}

This specification was found using the strategy pack "Point Placements" and has 572 rules.

Found on January 18, 2022.

Finding the specification took 62 seconds.

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\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) \\ F_{3} (x) & = F_{4} (x) F_{5} (x) \\ F_{4} (x) & = x \\ F_{5} (x) & = F_{29} (x) + F_{6} (x) \\ F_{6} (x) & = F_{1} (x) + F_{7} (x) \\ F_{7} (x) & = F_{8} (x) \\ F_{8} (x) & = F_{4} (x) F_{9} (x) \\ F_{9} (x) & = F_{10} (x) + F_{14} (x) \\ F_{10} (x) & = F_{1} (x) + F_{11} (x) \\ F_{11} (x) & = F_{12} (x) \\ F_{12} (x) & = F_{13} (x) F_{4} (x) \\ F_{13} (x) & = F_{1} (x) + F_{4} (x) \\ F_{14} (x) & = F_{15} (x) + F_{17} (x) \\ F_{15} (x) & = F_{16} (x) \\ F_{16} (x) & = F_{13} (x) F_{4} (x) \\ F_{17} (x) & = F_{18} (x) + F_{19} (x) + F_{24} (x) \\ F_{18} (x) & = 0 \\ F_{19} (x) & = F_{20} (x) F_{4} (x) \\ F_{20} (x) & = F_{11} (x) + F_{21} (x) \\ F_{21} (x) & = F_{22} (x) \\ F_{22} (x) & = F_{23} (x) F_{4} (x) \\ F_{23} (x) & = F_{4} (x) \\ F_{24} (x) & = F_{25} (x) F_{4} (x) \\ F_{25} (x) & = F_{15} (x) + F_{26} (x) \\ F_{26} (x) & = F_{27} (x) \\ F_{27} (x) & = F_{28} (x) F_{4} (x) \\ F_{28} (x) & = F_{4} (x) \\ F_{29} (x) & = F_{2} (x) + F_{30} (x) \\ F_{30} (x) & = F_{18} (x) + F_{31} (x) + F_{569} (x) \\ F_{31} (x) & = F_{32} (x) F_{4} (x) \\ F_{32} (x) & = F_{33} (x) + F_{54} (x) \\ F_{33} (x) & = F_{34} (x) + F_{7} (x) \\ F_{34} (x) & = F_{18} (x) + F_{35} (x) + F_{45} (x) \\ F_{35} (x) & = F_{36} (x) F_{4} (x) \\ F_{36} (x) & = F_{37} (x) + F_{39} (x) \\ F_{37} (x) & = F_{11} (x) + F_{38} (x) \\ F_{38} (x) & = F_{22} (x) \\ F_{39} (x) & = F_{17} (x) + F_{40} (x) \\ F_{40} (x) & = 2 F_{18} (x) + F_{41} (x) + F_{43} (x) \\ F_{41} (x) & = F_{4} (x) F_{42} (x) \\ F_{42} (x) & = F_{38} (x) \\ F_{43} (x) & = F_{4} (x) F_{44} (x) \\ F_{44} (x) & = F_{26} (x) \\ F_{45} (x) & = F_{4} (x) F_{46} (x) \\ F_{46} (x) & = F_{14} (x) + F_{47} (x) \\ F_{47} (x) & = F_{48} (x) + F_{49} (x) \\ F_{48} (x) & = F_{27} (x) \\ F_{49} (x) & = 2 F_{18} (x) + F_{50} (x) + F_{52} (x) \\ F_{50} (x) & = F_{4} (x) F_{51} (x) \\ F_{51} (x) & = F_{21} (x) \\ F_{52} (x) & = F_{4} (x) F_{53} (x) \\ F_{53} (x) & = F_{48} (x) \\ F_{54} (x) & = F_{30} (x) + F_{55} (x) \\ F_{55} (x) & = F_{18} (x) + F_{557} (x) + F_{56} (x) + F_{566} (x) \\ F_{56} (x) & = F_{4} (x) F_{57} (x) \\ F_{57} (x) & = F_{58} (x) + F_{69} (x) \\ F_{58} (x) & = F_{34} (x) + F_{59} (x) \\ F_{59} (x) & = F_{18} (x) + F_{60} (x) + F_{64} (x) + F_{67} (x) \\ F_{60} (x) & = F_{4} (x) F_{61} (x) \\ F_{61} (x) & = F_{62} (x) + F_{63} (x) \\ F_{62} (x) & = F_{38} (x) \\ F_{63} (x) & = F_{40} (x) \\ F_{64} (x) & = F_{4} (x) F_{65} (x) \\ F_{65} (x) & = F_{39} (x) + F_{66} (x) \\ F_{66} (x) & = F_{49} (x) \\ F_{67} (x) & = F_{4} (x) F_{68} (x) \\ F_{68} (x) & = F_{47} (x) \\ F_{69} (x) & = F_{55} (x) + F_{70} (x) \\ F_{70} (x) & = F_{127} (x) + F_{18} (x) + F_{502} (x) + F_{563} (x) + F_{71} (x) \\ F_{71} (x) & = F_{4} (x) F_{72} (x) \\ F_{72} (x) & = F_{73} (x) + F_{81} (x) \\ F_{73} (x) & = F_{59} (x) + F_{74} (x) \\ F_{74} (x) & = F_{18} (x) + F_{75} (x) + F_{76} (x) + F_{78} (x) + F_{80} (x) \\ F_{75} (x) & = 0 \\ F_{76} (x) & = F_{4} (x) F_{77} (x) \\ F_{77} (x) & = F_{63} (x) \\ F_{78} (x) & = F_{4} (x) F_{79} (x) \\ F_{79} (x) & = F_{66} (x) \\ F_{80} (x) & = 0 \\ F_{81} (x) & = F_{70} (x) + F_{82} (x) \\ F_{82} (x) & = F_{18} (x) + F_{426} (x) + F_{501} (x) + F_{83} (x) + F_{87} (x) + F_{88} (x) \\ F_{83} (x) & = F_{4} (x) F_{84} (x) \\ F_{84} (x) & = F_{85} (x) + F_{86} (x) \\ F_{85} (x) & = F_{74} (x) \\ F_{86} (x) & = F_{82} (x) \\ F_{87} (x) & = 0 \\ F_{88} (x) & = F_{4} (x) F_{89} (x) \\ F_{89} (x) & = F_{90} (x) \\ F_{90} (x) & = F_{91} (x) \\ F_{91} (x) & = 2 F_{18} (x) + F_{108} (x) + F_{130} (x) + F_{92} (x) \\ F_{92} (x) & = F_{4} (x) F_{93} (x) \\ F_{93} (x) & = F_{94} (x) + F_{98} (x) \\ F_{94} (x) & = F_{40} (x) + F_{95} (x) \\ F_{95} (x) & = F_{96} (x) \\ F_{96} (x) & = F_{4} (x) F_{97} (x) \\ F_{97} (x) & = F_{40} (x) \\ F_{98} (x) & = F_{91} (x) + F_{99} (x) \\ F_{99} (x) & = 2 F_{18} (x) + F_{100} (x) + F_{104} (x) + F_{105} (x) + F_{107} (x) \\ F_{100} (x) & = F_{101} (x) F_{4} (x) \\ F_{101} (x) & = F_{102} (x) + F_{103} (x) \\ F_{102} (x) & = F_{95} (x) \\ F_{103} (x) & = F_{99} (x) \\ F_{104} (x) & = 0 \\ F_{105} (x) & = F_{106} (x) F_{4} (x) \\ F_{106} (x) & = F_{91} (x) \\ F_{107} (x) & = 0 \\ F_{108} (x) & = F_{109} (x) F_{4} (x) \\ F_{109} (x) & = F_{110} (x) \\ F_{110} (x) & = 2 F_{18} (x) + F_{111} (x) + F_{424} (x) \\ F_{111} (x) & = F_{112} (x) F_{4} (x) \\ F_{112} (x) & = F_{113} (x) + F_{115} (x) \\ F_{113} (x) & = F_{114} (x) + F_{38} (x) \\ F_{114} (x) & = 2 F_{18} (x) + F_{43} (x) + F_{60} (x) \\ F_{115} (x) & = F_{110} (x) + F_{116} (x) \\ F_{116} (x) & = 2 F_{18} (x) + F_{117} (x) + F_{127} (x) + F_{130} (x) \\ F_{117} (x) & = F_{118} (x) F_{4} (x) \\ F_{118} (x) & = F_{119} (x) + F_{121} (x) \\ F_{119} (x) & = F_{114} (x) + F_{120} (x) \\ F_{120} (x) & = F_{76} (x) \\ F_{121} (x) & = F_{116} (x) + F_{122} (x) \\ F_{122} (x) & = 2 F_{18} (x) + F_{107} (x) + F_{123} (x) + F_{87} (x) + F_{88} (x) \\ F_{123} (x) & = F_{124} (x) F_{4} (x) \\ F_{124} (x) & = F_{125} (x) + F_{126} (x) \\ F_{125} (x) & = F_{120} (x) \\ F_{126} (x) & = F_{122} (x) \\ F_{127} (x) & = F_{128} (x) F_{4} (x) \\ F_{128} (x) & = F_{129} (x) + F_{90} (x) \\ F_{129} (x) & = F_{110} (x) \\ F_{130} (x) & = F_{131} (x) F_{4} (x) \\ F_{131} (x) & = F_{132} (x) \\ F_{132} (x) & = 2 F_{18} (x) + F_{133} (x) + F_{378} (x) \\ F_{133} (x) & = F_{134} (x) F_{4} (x) \\ F_{134} (x) & = F_{135} (x) + F_{145} (x) \\ F_{135} (x) & = F_{136} (x) + F_{26} (x) \\ F_{136} (x) & = F_{137} (x) \\ F_{137} (x) & = F_{138} (x) F_{4} (x) \\ F_{138} (x) & = F_{139} (x) \\ F_{139} (x) & = F_{140} (x) + F_{144} (x) + F_{18} (x) \\ F_{140} (x) & = F_{141} (x) F_{4} (x) \\ F_{141} (x) & = F_{142} (x) + F_{4} (x) \\ F_{142} (x) & = F_{143} (x) \\ F_{143} (x) & = x^{2} \\ F_{144} (x) & = F_{15} (x) F_{4} (x) \\ F_{145} (x) & = F_{132} (x) + F_{146} (x) \\ F_{146} (x) & = 2 F_{18} (x) + F_{147} (x) + F_{375} (x) + F_{376} (x) \\ F_{147} (x) & = F_{148} (x) F_{4} (x) \\ F_{148} (x) & = F_{149} (x) + F_{157} (x) \\ F_{149} (x) & = F_{136} (x) + F_{150} (x) \\ F_{150} (x) & = F_{151} (x) \\ F_{151} (x) & = F_{152} (x) F_{4} (x) \\ F_{152} (x) & = F_{153} (x) \\ F_{153} (x) & = 2 F_{18} (x) + F_{154} (x) + F_{156} (x) \\ F_{154} (x) & = F_{155} (x) F_{4} (x) \\ F_{155} (x) & = F_{142} (x) \\ F_{156} (x) & = F_{4} (x) F_{48} (x) \\ F_{157} (x) & = F_{146} (x) + F_{158} (x) \\ F_{158} (x) & = 2 F_{18} (x) + F_{159} (x) + F_{163} (x) + F_{164} (x) + F_{165} (x) \\ F_{159} (x) & = F_{160} (x) F_{4} (x) \\ F_{160} (x) & = F_{161} (x) + F_{162} (x) \\ F_{161} (x) & = F_{150} (x) \\ F_{162} (x) & = F_{158} (x) \\ F_{163} (x) & = 0 \\ F_{164} (x) & = 0 \\ F_{165} (x) & = F_{166} (x) F_{4} (x) \\ F_{166} (x) & = F_{167} (x) \\ F_{167} (x) & = 2 F_{18} (x) + F_{168} (x) + F_{357} (x) + F_{422} (x) \\ F_{168} (x) & = F_{169} (x) F_{4} (x) \\ F_{169} (x) & = F_{170} (x) + F_{178} (x) \\ F_{170} (x) & = F_{153} (x) + F_{171} (x) \\ F_{171} (x) & = F_{172} (x) \\ F_{172} (x) & = F_{173} (x) F_{4} (x) \\ F_{173} (x) & = F_{174} (x) \\ F_{174} (x) & = 2 F_{18} (x) + F_{175} (x) + F_{177} (x) \\ F_{175} (x) & = F_{176} (x) F_{4} (x) \\ F_{176} (x) & = F_{142} (x) \\ F_{177} (x) & = F_{26} (x) F_{4} (x) \\ F_{178} (x) & = F_{167} (x) + F_{179} (x) \\ F_{179} (x) & = 2 F_{18} (x) + F_{180} (x) + F_{184} (x) + F_{185} (x) + F_{327} (x) \\ F_{180} (x) & = F_{181} (x) F_{4} (x) \\ F_{181} (x) & = F_{182} (x) + F_{183} (x) \\ F_{182} (x) & = F_{171} (x) \\ F_{183} (x) & = F_{179} (x) \\ F_{184} (x) & = 0 \\ F_{185} (x) & = F_{186} (x) F_{4} (x) \\ F_{186} (x) & = F_{187} (x) \\ F_{187} (x) & = 2 F_{18} (x) + F_{188} (x) + F_{204} (x) + F_{236} (x) \\ F_{188} (x) & = F_{189} (x) F_{4} (x) \\ F_{189} (x) & = F_{190} (x) + F_{194} (x) \\ F_{190} (x) & = F_{174} (x) + F_{191} (x) \\ F_{191} (x) & = F_{192} (x) \\ F_{192} (x) & = F_{193} (x) F_{4} (x) \\ F_{193} (x) & = F_{174} (x) \\ F_{194} (x) & = F_{187} (x) + F_{195} (x) \\ F_{195} (x) & = 2 F_{18} (x) + F_{196} (x) + F_{200} (x) + F_{201} (x) + F_{203} (x) \\ F_{196} (x) & = F_{197} (x) F_{4} (x) \\ F_{197} (x) & = F_{198} (x) + F_{199} (x) \\ F_{198} (x) & = F_{191} (x) \\ F_{199} (x) & = F_{195} (x) \\ F_{200} (x) & = 0 \\ F_{201} (x) & = F_{202} (x) F_{4} (x) \\ F_{202} (x) & = F_{187} (x) \\ F_{203} (x) & = 0 \\ F_{204} (x) & = F_{205} (x) F_{4} (x) \\ F_{205} (x) & = F_{206} (x) \\ F_{206} (x) & = 2 F_{18} (x) + F_{207} (x) + F_{237} (x) \\ F_{207} (x) & = F_{208} (x) F_{4} (x) \\ F_{208} (x) & = F_{209} (x) + F_{215} (x) \\ F_{209} (x) & = F_{142} (x) + F_{210} (x) \\ F_{210} (x) & = 2 F_{18} (x) + F_{177} (x) + F_{211} (x) \\ F_{211} (x) & = F_{212} (x) F_{4} (x) \\ F_{212} (x) & = F_{213} (x) + F_{214} (x) \\ F_{213} (x) & = F_{142} (x) \\ F_{214} (x) & = F_{174} (x) \\ F_{215} (x) & = F_{206} (x) + F_{216} (x) \\ F_{216} (x) & = 2 F_{18} (x) + F_{217} (x) + F_{233} (x) + F_{236} (x) \\ F_{217} (x) & = F_{218} (x) F_{4} (x) \\ F_{218} (x) & = F_{219} (x) + F_{223} (x) \\ F_{219} (x) & = F_{210} (x) + F_{220} (x) \\ F_{220} (x) & = F_{221} (x) \\ F_{221} (x) & = F_{222} (x) F_{4} (x) \\ F_{222} (x) & = F_{214} (x) \\ F_{223} (x) & = F_{216} (x) + F_{224} (x) \\ F_{224} (x) & = 2 F_{18} (x) + F_{203} (x) + F_{225} (x) + F_{229} (x) + F_{230} (x) \\ F_{225} (x) & = F_{226} (x) F_{4} (x) \\ F_{226} (x) & = F_{227} (x) + F_{228} (x) \\ F_{227} (x) & = F_{220} (x) \\ F_{228} (x) & = F_{224} (x) \\ F_{229} (x) & = 0 \\ F_{230} (x) & = F_{231} (x) F_{4} (x) \\ F_{231} (x) & = F_{232} (x) \\ F_{232} (x) & = F_{187} (x) \\ F_{233} (x) & = F_{234} (x) F_{4} (x) \\ F_{234} (x) & = F_{232} (x) + F_{235} (x) \\ F_{235} (x) & = F_{206} (x) \\ F_{236} (x) & = F_{132} (x) F_{4} (x) \\ F_{237} (x) & = F_{238} (x) F_{4} (x) \\ F_{238} (x) & = F_{18} (x) + F_{239} (x) + F_{421} (x) \\ F_{239} (x) & = F_{240} (x) F_{4} (x) \\ F_{240} (x) & = F_{241} (x) + F_{251} (x) \\ F_{241} (x) & = F_{242} (x) + F_{4} (x) \\ F_{242} (x) & = F_{144} (x) + F_{18} (x) + F_{243} (x) \\ F_{243} (x) & = F_{244} (x) F_{4} (x) \\ F_{244} (x) & = F_{245} (x) + F_{247} (x) \\ F_{245} (x) & = F_{246} (x) + F_{4} (x) \\ F_{246} (x) & = F_{143} (x) \\ F_{247} (x) & = F_{139} (x) + F_{248} (x) \\ F_{248} (x) & = 2 F_{18} (x) + F_{177} (x) + F_{249} (x) \\ F_{249} (x) & = F_{250} (x) F_{4} (x) \\ F_{250} (x) & = F_{246} (x) \\ F_{251} (x) & = F_{238} (x) + F_{252} (x) \\ F_{252} (x) & = F_{18} (x) + F_{253} (x) + F_{382} (x) + F_{414} (x) \\ F_{253} (x) & = F_{254} (x) F_{4} (x) \\ F_{254} (x) & = F_{255} (x) + F_{264} (x) \\ F_{255} (x) & = F_{242} (x) + F_{256} (x) \\ F_{256} (x) & = F_{156} (x) + F_{18} (x) + F_{257} (x) + F_{261} (x) \\ F_{257} (x) & = F_{258} (x) F_{4} (x) \\ F_{258} (x) & = F_{259} (x) + F_{260} (x) \\ F_{259} (x) & = F_{246} (x) \\ F_{260} (x) & = F_{248} (x) \\ F_{261} (x) & = F_{262} (x) F_{4} (x) \\ F_{262} (x) & = F_{247} (x) + F_{263} (x) \\ F_{263} (x) & = F_{153} (x) \\ F_{264} (x) & = F_{252} (x) + F_{265} (x) \\ F_{265} (x) & = F_{18} (x) + F_{266} (x) + F_{321} (x) + F_{328} (x) + F_{357} (x) \\ F_{266} (x) & = F_{267} (x) F_{4} (x) \\ F_{267} (x) & = F_{268} (x) + F_{276} (x) \\ F_{268} (x) & = F_{256} (x) + F_{269} (x) \\ F_{269} (x) & = F_{18} (x) + F_{270} (x) + F_{271} (x) + F_{273} (x) + F_{275} (x) \\ F_{270} (x) & = 0 \\ F_{271} (x) & = F_{272} (x) F_{4} (x) \\ F_{272} (x) & = F_{260} (x) \\ F_{273} (x) & = F_{274} (x) F_{4} (x) \\ F_{274} (x) & = F_{263} (x) \\ F_{275} (x) & = 0 \\ F_{276} (x) & = F_{265} (x) + F_{277} (x) \\ F_{277} (x) & = F_{18} (x) + F_{278} (x) + F_{282} (x) + F_{283} (x) + F_{324} (x) + F_{327} (x) \\ F_{278} (x) & = F_{279} (x) F_{4} (x) \\ F_{279} (x) & = F_{280} (x) + F_{281} (x) \\ F_{280} (x) & = F_{269} (x) \\ F_{281} (x) & = F_{277} (x) \\ F_{282} (x) & = 0 \\ F_{283} (x) & = F_{284} (x) F_{4} (x) \\ F_{284} (x) & = F_{285} (x) \\ F_{285} (x) & = F_{286} (x) \\ F_{286} (x) & = 2 F_{18} (x) + F_{236} (x) + F_{287} (x) + F_{302} (x) \\ F_{287} (x) & = F_{288} (x) F_{4} (x) \\ F_{288} (x) & = F_{289} (x) + F_{293} (x) \\ F_{289} (x) & = F_{248} (x) + F_{290} (x) \\ F_{290} (x) & = F_{291} (x) \\ F_{291} (x) & = F_{292} (x) F_{4} (x) \\ F_{292} (x) & = F_{248} (x) \\ F_{293} (x) & = F_{286} (x) + F_{294} (x) \\ F_{294} (x) & = 2 F_{18} (x) + F_{203} (x) + F_{295} (x) + F_{299} (x) + F_{300} (x) \\ F_{295} (x) & = F_{296} (x) F_{4} (x) \\ F_{296} (x) & = F_{297} (x) + F_{298} (x) \\ F_{297} (x) & = F_{290} (x) \\ F_{298} (x) & = F_{294} (x) \\ F_{299} (x) & = 0 \\ F_{300} (x) & = F_{301} (x) F_{4} (x) \\ F_{301} (x) & = F_{286} (x) \\ F_{302} (x) & = F_{303} (x) F_{4} (x) \\ F_{303} (x) & = F_{304} (x) \\ F_{304} (x) & = 2 F_{18} (x) + F_{237} (x) + F_{305} (x) \\ F_{305} (x) & = F_{306} (x) F_{4} (x) \\ F_{306} (x) & = F_{307} (x) + F_{309} (x) \\ F_{307} (x) & = F_{246} (x) + F_{308} (x) \\ F_{308} (x) & = 2 F_{18} (x) + F_{177} (x) + F_{257} (x) \\ F_{309} (x) & = F_{304} (x) + F_{310} (x) \\ F_{310} (x) & = 2 F_{18} (x) + F_{236} (x) + F_{311} (x) + F_{321} (x) \\ F_{311} (x) & = F_{312} (x) F_{4} (x) \\ F_{312} (x) & = F_{313} (x) + F_{315} (x) \\ F_{313} (x) & = F_{308} (x) + F_{314} (x) \\ F_{314} (x) & = F_{271} (x) \\ F_{315} (x) & = F_{310} (x) + F_{316} (x) \\ F_{316} (x) & = 2 F_{18} (x) + F_{203} (x) + F_{282} (x) + F_{283} (x) + F_{317} (x) \\ F_{317} (x) & = F_{318} (x) F_{4} (x) \\ F_{318} (x) & = F_{319} (x) + F_{320} (x) \\ F_{319} (x) & = F_{314} (x) \\ F_{320} (x) & = F_{316} (x) \\ F_{321} (x) & = F_{322} (x) F_{4} (x) \\ F_{322} (x) & = F_{285} (x) + F_{323} (x) \\ F_{323} (x) & = F_{304} (x) \\ F_{324} (x) & = F_{325} (x) F_{4} (x) \\ F_{325} (x) & = F_{326} (x) \\ F_{326} (x) & = F_{167} (x) \\ F_{327} (x) & = 0 \\ F_{328} (x) & = F_{329} (x) F_{4} (x) \\ F_{329} (x) & = F_{326} (x) + F_{330} (x) \\ F_{330} (x) & = F_{286} (x) + F_{331} (x) \\ F_{331} (x) & = F_{18} (x) + F_{332} (x) + F_{380} (x) + F_{382} (x) \\ F_{332} (x) & = F_{333} (x) F_{4} (x) \\ F_{333} (x) & = F_{334} (x) + F_{338} (x) \\ F_{334} (x) & = F_{139} (x) + F_{335} (x) \\ F_{335} (x) & = F_{156} (x) + F_{18} (x) + F_{211} (x) + F_{336} (x) \\ F_{336} (x) & = F_{337} (x) F_{4} (x) \\ F_{337} (x) & = F_{139} (x) + F_{174} (x) \\ F_{338} (x) & = F_{331} (x) + F_{339} (x) \\ F_{339} (x) & = F_{18} (x) + F_{233} (x) + F_{340} (x) + F_{355} (x) + F_{357} (x) \\ F_{340} (x) & = F_{341} (x) F_{4} (x) \\ F_{341} (x) & = F_{342} (x) + F_{347} (x) \\ F_{342} (x) & = F_{335} (x) + F_{343} (x) \\ F_{343} (x) & = F_{18} (x) + F_{221} (x) + F_{275} (x) + F_{344} (x) + F_{345} (x) \\ F_{344} (x) & = 0 \\ F_{345} (x) & = F_{346} (x) F_{4} (x) \\ F_{346} (x) & = F_{153} (x) \\ F_{347} (x) & = F_{339} (x) + F_{348} (x) \\ F_{348} (x) & = F_{18} (x) + F_{229} (x) + F_{230} (x) + F_{327} (x) + F_{349} (x) + F_{353} (x) \\ F_{349} (x) & = F_{350} (x) F_{4} (x) \\ F_{350} (x) & = F_{351} (x) + F_{352} (x) \\ F_{351} (x) & = F_{343} (x) \\ F_{352} (x) & = F_{348} (x) \\ F_{353} (x) & = F_{354} (x) F_{4} (x) \\ F_{354} (x) & = F_{167} (x) \\ F_{355} (x) & = F_{356} (x) F_{4} (x) \\ F_{356} (x) & = F_{187} (x) + F_{331} (x) \\ F_{357} (x) & = F_{358} (x) F_{4} (x) \\ F_{358} (x) & = 2 F_{18} (x) + F_{359} (x) + F_{378} (x) \\ F_{359} (x) & = F_{360} (x) F_{4} (x) \\ F_{360} (x) & = F_{361} (x) + F_{363} (x) \\ F_{361} (x) & = F_{362} (x) + F_{48} (x) \\ F_{362} (x) & = F_{137} (x) \\ F_{363} (x) & = F_{358} (x) + F_{364} (x) \\ F_{364} (x) & = 2 F_{18} (x) + F_{365} (x) + F_{375} (x) + F_{376} (x) \\ F_{365} (x) & = F_{366} (x) F_{4} (x) \\ F_{366} (x) & = F_{367} (x) + F_{369} (x) \\ F_{367} (x) & = F_{362} (x) + F_{368} (x) \\ F_{368} (x) & = F_{151} (x) \\ F_{369} (x) & = F_{364} (x) + F_{370} (x) \\ F_{370} (x) & = 2 F_{18} (x) + F_{163} (x) + F_{164} (x) + F_{165} (x) + F_{371} (x) \\ F_{371} (x) & = F_{372} (x) F_{4} (x) \\ F_{372} (x) & = F_{373} (x) + F_{374} (x) \\ F_{373} (x) & = F_{368} (x) \\ F_{374} (x) & = F_{370} (x) \\ F_{375} (x) & = 0 \\ F_{376} (x) & = F_{377} (x) F_{4} (x) \\ F_{377} (x) & = F_{331} (x) \\ F_{378} (x) & = F_{379} (x) F_{4} (x) \\ F_{379} (x) & = F_{238} (x) \\ F_{380} (x) & = F_{381} (x) F_{4} (x) \\ F_{381} (x) & = F_{206} (x) + F_{238} (x) \\ F_{382} (x) & = F_{383} (x) F_{4} (x) \\ F_{383} (x) & = F_{18} (x) + F_{384} (x) + F_{419} (x) \\ F_{384} (x) & = F_{385} (x) F_{4} (x) \\ F_{385} (x) & = F_{386} (x) + F_{390} (x) \\ F_{386} (x) & = F_{15} (x) + F_{387} (x) \\ F_{387} (x) & = F_{18} (x) + F_{243} (x) + F_{388} (x) \\ F_{388} (x) & = F_{389} (x) F_{4} (x) \\ F_{389} (x) & = F_{139} (x) + F_{15} (x) \\ F_{390} (x) & = F_{383} (x) + F_{391} (x) \\ F_{391} (x) & = F_{18} (x) + F_{392} (x) + F_{414} (x) + F_{417} (x) \\ F_{392} (x) & = F_{393} (x) F_{4} (x) \\ F_{393} (x) & = F_{394} (x) + F_{398} (x) \\ F_{394} (x) & = F_{387} (x) + F_{395} (x) \\ F_{395} (x) & = F_{18} (x) + F_{257} (x) + F_{261} (x) + F_{396} (x) \\ F_{396} (x) & = F_{397} (x) F_{4} (x) \\ F_{397} (x) & = F_{153} (x) + F_{48} (x) \\ F_{398} (x) & = F_{391} (x) + F_{399} (x) \\ F_{399} (x) & = F_{18} (x) + F_{321} (x) + F_{328} (x) + F_{400} (x) + F_{412} (x) \\ F_{400} (x) & = F_{4} (x) F_{401} (x) \\ F_{401} (x) & = F_{402} (x) + F_{405} (x) \\ F_{402} (x) & = F_{395} (x) + F_{403} (x) \\ F_{403} (x) & = F_{18} (x) + F_{270} (x) + F_{271} (x) + F_{273} (x) + F_{404} (x) \\ F_{404} (x) & = 0 \\ F_{405} (x) & = F_{399} (x) + F_{406} (x) \\ F_{406} (x) & = F_{18} (x) + F_{282} (x) + F_{283} (x) + F_{324} (x) + F_{407} (x) + F_{411} (x) \\ F_{407} (x) & = F_{4} (x) F_{408} (x) \\ F_{408} (x) & = F_{409} (x) + F_{410} (x) \\ F_{409} (x) & = F_{403} (x) \\ F_{410} (x) & = F_{406} (x) \\ F_{411} (x) & = 0 \\ F_{412} (x) & = F_{4} (x) F_{413} (x) \\ F_{413} (x) & = F_{167} (x) + F_{358} (x) \\ F_{414} (x) & = F_{4} (x) F_{415} (x) \\ F_{415} (x) & = F_{330} (x) + F_{416} (x) \\ F_{416} (x) & = F_{238} (x) + F_{304} (x) \\ F_{417} (x) & = F_{4} (x) F_{418} (x) \\ F_{418} (x) & = F_{331} (x) + F_{383} (x) \\ F_{419} (x) & = F_{4} (x) F_{420} (x) \\ F_{420} (x) & = F_{2} (x) + F_{238} (x) \\ F_{421} (x) & = F_{2} (x) F_{4} (x) \\ F_{422} (x) & = F_{4} (x) F_{423} (x) \\ F_{423} (x) & = F_{206} (x) \\ F_{424} (x) & = F_{4} (x) F_{425} (x) \\ F_{425} (x) & = F_{238} (x) \\ F_{426} (x) & = F_{4} (x) F_{427} (x) \\ F_{427} (x) & = F_{428} (x) \\ F_{428} (x) & = F_{429} (x) \\ F_{429} (x) & = 2 F_{18} (x) + F_{430} (x) + F_{497} (x) + F_{499} (x) \\ F_{430} (x) & = F_{4} (x) F_{431} (x) \\ F_{431} (x) & = F_{432} (x) + F_{439} (x) \\ F_{432} (x) & = F_{433} (x) + F_{49} (x) \\ F_{433} (x) & = F_{434} (x) \\ F_{434} (x) & = F_{4} (x) F_{435} (x) \\ F_{435} (x) & = F_{436} (x) \\ F_{436} (x) & = 2 F_{18} (x) + F_{43} (x) + F_{437} (x) \\ F_{437} (x) & = F_{4} (x) F_{438} (x) \\ F_{438} (x) & = F_{21} (x) \\ F_{439} (x) & = F_{429} (x) + F_{440} (x) \\ F_{440} (x) & = 2 F_{18} (x) + F_{441} (x) + F_{445} (x) + F_{446} (x) + F_{496} (x) \\ F_{441} (x) & = F_{4} (x) F_{442} (x) \\ F_{442} (x) & = F_{443} (x) + F_{444} (x) \\ F_{443} (x) & = F_{433} (x) \\ F_{444} (x) & = F_{440} (x) \\ F_{445} (x) & = 0 \\ F_{446} (x) & = F_{4} (x) F_{447} (x) \\ F_{447} (x) & = F_{448} (x) \\ F_{448} (x) & = 2 F_{18} (x) + F_{130} (x) + F_{449} (x) + F_{464} (x) \\ F_{449} (x) & = F_{4} (x) F_{450} (x) \\ F_{450} (x) & = F_{451} (x) + F_{455} (x) \\ F_{451} (x) & = F_{436} (x) + F_{452} (x) \\ F_{452} (x) & = F_{453} (x) \\ F_{453} (x) & = F_{4} (x) F_{454} (x) \\ F_{454} (x) & = F_{436} (x) \\ F_{455} (x) & = F_{448} (x) + F_{456} (x) \\ F_{456} (x) & = 2 F_{18} (x) + F_{107} (x) + F_{457} (x) + F_{461} (x) + F_{462} (x) \\ F_{457} (x) & = F_{4} (x) F_{458} (x) \\ F_{458} (x) & = F_{459} (x) + F_{460} (x) \\ F_{459} (x) & = F_{452} (x) \\ F_{460} (x) & = F_{456} (x) \\ F_{461} (x) & = 0 \\ F_{462} (x) & = F_{4} (x) F_{463} (x) \\ F_{463} (x) & = F_{448} (x) \\ F_{464} (x) & = F_{4} (x) F_{465} (x) \\ F_{465} (x) & = F_{466} (x) \\ F_{466} (x) & = 2 F_{18} (x) + F_{424} (x) + F_{467} (x) \\ F_{467} (x) & = F_{4} (x) F_{468} (x) \\ F_{468} (x) & = F_{469} (x) + F_{475} (x) \\ F_{469} (x) & = F_{21} (x) + F_{470} (x) \\ F_{470} (x) & = 2 F_{18} (x) + F_{43} (x) + F_{471} (x) \\ F_{471} (x) & = F_{4} (x) F_{472} (x) \\ F_{472} (x) & = F_{473} (x) + F_{474} (x) \\ F_{473} (x) & = F_{21} (x) \\ F_{474} (x) & = F_{436} (x) \\ F_{475} (x) & = F_{466} (x) + F_{476} (x) \\ F_{476} (x) & = 2 F_{18} (x) + F_{130} (x) + F_{477} (x) + F_{493} (x) \\ F_{477} (x) & = F_{4} (x) F_{478} (x) \\ F_{478} (x) & = F_{479} (x) + F_{483} (x) \\ F_{479} (x) & = F_{470} (x) + F_{480} (x) \\ F_{480} (x) & = F_{481} (x) \\ F_{481} (x) & = F_{4} (x) F_{482} (x) \\ F_{482} (x) & = F_{474} (x) \\ F_{483} (x) & = F_{476} (x) + F_{484} (x) \\ F_{484} (x) & = 2 F_{18} (x) + F_{107} (x) + F_{485} (x) + F_{489} (x) + F_{490} (x) \\ F_{485} (x) & = F_{4} (x) F_{486} (x) \\ F_{486} (x) & = F_{487} (x) + F_{488} (x) \\ F_{487} (x) & = F_{480} (x) \\ F_{488} (x) & = F_{484} (x) \\ F_{489} (x) & = 0 \\ F_{490} (x) & = F_{4} (x) F_{491} (x) \\ F_{491} (x) & = F_{492} (x) \\ F_{492} (x) & = F_{448} (x) \\ F_{493} (x) & = F_{4} (x) F_{494} (x) \\ F_{494} (x) & = F_{492} (x) + F_{495} (x) \\ F_{495} (x) & = F_{466} (x) \\ F_{496} (x) & = 0 \\ F_{497} (x) & = F_{4} (x) F_{498} (x) \\ F_{498} (x) & = F_{466} (x) \\ F_{499} (x) & = F_{4} (x) F_{500} (x) \\ F_{500} (x) & = F_{358} (x) \\ F_{501} (x) & = 0 \\ F_{502} (x) & = F_{4} (x) F_{503} (x) \\ F_{503} (x) & = F_{428} (x) + F_{504} (x) \\ F_{504} (x) & = F_{505} (x) + F_{91} (x) \\ F_{505} (x) & = F_{18} (x) + F_{506} (x) + F_{532} (x) + F_{560} (x) \\ F_{506} (x) & = F_{4} (x) F_{507} (x) \\ F_{507} (x) & = F_{508} (x) + F_{512} (x) \\ F_{508} (x) & = F_{17} (x) + F_{509} (x) \\ F_{509} (x) & = F_{18} (x) + F_{471} (x) + F_{510} (x) + F_{52} (x) \\ F_{510} (x) & = F_{4} (x) F_{511} (x) \\ F_{511} (x) & = F_{17} (x) + F_{436} (x) \\ F_{512} (x) & = F_{505} (x) + F_{513} (x) \\ F_{513} (x) & = F_{18} (x) + F_{493} (x) + F_{499} (x) + F_{514} (x) + F_{530} (x) \\ F_{514} (x) & = F_{4} (x) F_{515} (x) \\ F_{515} (x) & = F_{516} (x) + F_{522} (x) \\ F_{516} (x) & = F_{509} (x) + F_{517} (x) \\ F_{517} (x) & = F_{18} (x) + F_{481} (x) + F_{518} (x) + F_{519} (x) + F_{521} (x) \\ F_{518} (x) & = 0 \\ F_{519} (x) & = F_{4} (x) F_{520} (x) \\ F_{520} (x) & = F_{49} (x) \\ F_{521} (x) & = 0 \\ F_{522} (x) & = F_{513} (x) + F_{523} (x) \\ F_{523} (x) & = F_{18} (x) + F_{489} (x) + F_{490} (x) + F_{496} (x) + F_{524} (x) + F_{528} (x) \\ F_{524} (x) & = F_{4} (x) F_{525} (x) \\ F_{525} (x) & = F_{526} (x) + F_{527} (x) \\ F_{526} (x) & = F_{517} (x) \\ F_{527} (x) & = F_{523} (x) \\ F_{528} (x) & = F_{4} (x) F_{529} (x) \\ F_{529} (x) & = F_{429} (x) \\ F_{530} (x) & = F_{4} (x) F_{531} (x) \\ F_{531} (x) & = F_{448} (x) + F_{505} (x) \\ F_{532} (x) & = F_{4} (x) F_{533} (x) \\ F_{533} (x) & = F_{466} (x) + F_{534} (x) \\ F_{534} (x) & = F_{18} (x) + F_{535} (x) + F_{562} (x) \\ F_{535} (x) & = F_{4} (x) F_{536} (x) \\ F_{536} (x) & = F_{537} (x) + F_{539} (x) \\ F_{537} (x) & = F_{11} (x) + F_{538} (x) \\ F_{538} (x) & = F_{18} (x) + F_{24} (x) + F_{35} (x) \\ F_{539} (x) & = F_{534} (x) + F_{540} (x) \\ F_{540} (x) & = F_{18} (x) + F_{541} (x) + F_{557} (x) + F_{560} (x) \\ F_{541} (x) & = F_{4} (x) F_{542} (x) \\ F_{542} (x) & = F_{543} (x) + F_{545} (x) \\ F_{543} (x) & = F_{538} (x) + F_{544} (x) \\ F_{544} (x) & = F_{18} (x) + F_{52} (x) + F_{60} (x) + F_{64} (x) \\ F_{545} (x) & = F_{540} (x) + F_{546} (x) \\ F_{546} (x) & = F_{127} (x) + F_{18} (x) + F_{499} (x) + F_{502} (x) + F_{547} (x) \\ F_{547} (x) & = F_{4} (x) F_{548} (x) \\ F_{548} (x) & = F_{549} (x) + F_{551} (x) \\ F_{549} (x) & = F_{544} (x) + F_{550} (x) \\ F_{550} (x) & = F_{18} (x) + F_{521} (x) + F_{75} (x) + F_{76} (x) + F_{78} (x) \\ F_{551} (x) & = F_{546} (x) + F_{552} (x) \\ F_{552} (x) & = F_{18} (x) + F_{426} (x) + F_{496} (x) + F_{553} (x) + F_{87} (x) + F_{88} (x) \\ F_{553} (x) & = F_{4} (x) F_{554} (x) \\ F_{554} (x) & = F_{555} (x) + F_{556} (x) \\ F_{555} (x) & = F_{550} (x) \\ F_{556} (x) & = F_{552} (x) \\ F_{557} (x) & = F_{4} (x) F_{558} (x) \\ F_{558} (x) & = F_{504} (x) + F_{559} (x) \\ F_{559} (x) & = F_{110} (x) + F_{534} (x) \\ F_{560} (x) & = F_{4} (x) F_{561} (x) \\ F_{561} (x) & = F_{132} (x) + F_{383} (x) \\ F_{562} (x) & = F_{4} (x) F_{420} (x) \\ F_{563} (x) & = F_{4} (x) F_{564} (x) \\ F_{564} (x) & = F_{565} (x) \\ F_{565} (x) & = F_{358} (x) + F_{429} (x) \\ F_{566} (x) & = F_{4} (x) F_{567} (x) \\ F_{567} (x) & = F_{565} (x) + F_{568} (x) \\ F_{568} (x) & = F_{383} (x) + F_{505} (x) \\ F_{569} (x) & = F_{4} (x) F_{570} (x) \\ F_{570} (x) & = F_{568} (x) + F_{571} (x) \\ F_{571} (x) & = F_{2} (x) + F_{534} (x) \end{aligned}

Av(1234, 1432)

This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 160 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point Placements" and has 572 rules.

Proof Tree

System of Equations