PermPal Database

Av(1243, 2143, 2341)

Generating Function

\frac{x^{4} - 3 x^{3} + 10 x^{2} - 6 x + 1}{(x^{2} - 3 x + 1) (2 x^{2} - 4 x + 1)}

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Counting Sequence

1, 1, 2, 6, 21, 75, 266, 935, 3263, 11326, 39155, 134955, 464094, 1593231, 5462447, ...

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Implicit Equation for the Generating Function

(x^{2} - 3 x + 1) (2 x^{2} - 4 x + 1) F (x) - x^{4} + 3 x^{3} - 10 x^{2} + 6 x - 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (4) = 21

a (n + 4) = - 2 a (n) + 10 a (n + 1) - 15 a (n + 2) + 7 a (n + 3), n \geq 5

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Explicit Closed Form

{\begin{array}{cc} 1 & n = 0 \\ \frac{(8 \sqrt{5} - 20) {(\frac{3}{2} - \frac{\sqrt{5}}{2})}^{- n}}{20} + \frac{(- 8 \sqrt{5} - 20) {(\frac{3}{2} + \frac{\sqrt{5}}{2})}^{- n}}{20} + \\ \frac{(- 15 \sqrt{2} + 25) {(1 - \frac{\sqrt{2}}{2})}^{- n}}{20} + \frac{(15 \sqrt{2} + 25) {(1 + \frac{\sqrt{2}}{2})}^{- n}}{20} & otherwise \end{array}

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This specification was found using the strategy pack "Point Placements" and has 114 rules.

Found on January 18, 2022.

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_{12} (x) F_{4} (x) \\ F_{4} (x) & = F_{40} (x) + F_{5} (x) \\ F_{5} (x) & = F_{1} (x) + F_{6} (x) \\ F_{6} (x) & = F_{7} (x) \\ F_{7} (x) & = F_{12} (x) F_{8} (x) \\ F_{8} (x) & = F_{13} (x) + F_{9} (x) \\ F_{9} (x) & = F_{1} (x) + F_{10} (x) \\ F_{10} (x) & = F_{11} (x) \\ F_{11} (x) & = F_{12} (x) F_{9} (x) \\ F_{12} (x) & = x \\ F_{13} (x) & = F_{14} (x) + F_{26} (x) \\ F_{14} (x) & = F_{15} (x) \\ F_{15} (x) & = F_{12} (x) F_{16} (x) \\ F_{16} (x) & = F_{17} (x) + F_{18} (x) \\ F_{17} (x) & = F_{1} (x) + F_{12} (x) \\ F_{18} (x) & = F_{14} (x) + F_{19} (x) \\ F_{19} (x) & = F_{20} (x) + F_{21} (x) + F_{25} (x) \\ F_{20} (x) & = 0 \\ F_{21} (x) & = F_{12} (x) F_{22} (x) \\ F_{22} (x) & = F_{23} (x) + F_{24} (x) \\ F_{23} (x) & = F_{12} (x) \\ F_{24} (x) & = F_{19} (x) \\ F_{25} (x) & = F_{12} (x) F_{14} (x) \\ F_{26} (x) & = F_{20} (x) + F_{27} (x) + F_{39} (x) \\ F_{27} (x) & = F_{12} (x) F_{28} (x) \\ F_{28} (x) & = F_{29} (x) + F_{32} (x) \\ F_{29} (x) & = F_{10} (x) + F_{30} (x) \\ F_{30} (x) & = F_{31} (x) \\ F_{31} (x) & = F_{10} (x) F_{12} (x) \\ F_{32} (x) & = F_{26} (x) + F_{33} (x) \\ F_{33} (x) & = 2 F_{20} (x) + F_{34} (x) + F_{38} (x) \\ F_{34} (x) & = F_{12} (x) F_{35} (x) \\ F_{35} (x) & = F_{36} (x) + F_{37} (x) \\ F_{36} (x) & = F_{30} (x) \\ F_{37} (x) & = F_{33} (x) \\ F_{38} (x) & = F_{12} (x) F_{26} (x) \\ F_{39} (x) & = F_{12} (x) F_{13} (x) \\ F_{40} (x) & = F_{41} (x) + F_{60} (x) \\ F_{41} (x) & = F_{42} (x) \\ F_{42} (x) & = F_{12} (x) F_{43} (x) \\ F_{43} (x) & = F_{44} (x) + F_{47} (x) \\ F_{44} (x) & = F_{1} (x) + F_{45} (x) \\ F_{45} (x) & = F_{46} (x) \\ F_{46} (x) & = F_{12} (x) F_{44} (x) \\ F_{47} (x) & = F_{41} (x) + F_{48} (x) \\ F_{48} (x) & = F_{20} (x) + F_{49} (x) + F_{59} (x) \\ F_{49} (x) & = F_{12} (x) F_{50} (x) \\ F_{50} (x) & = F_{51} (x) + F_{54} (x) \\ F_{51} (x) & = F_{12} (x) + F_{52} (x) \\ F_{52} (x) & = F_{53} (x) \\ F_{53} (x) & = F_{12} (x) F_{45} (x) \\ F_{54} (x) & = F_{55} (x) + F_{57} (x) \\ F_{55} (x) & = F_{20} (x) + F_{49} (x) + F_{56} (x) \\ F_{56} (x) & = F_{12} (x) F_{41} (x) \\ F_{57} (x) & = F_{58} (x) \\ F_{58} (x) & = F_{12} (x) F_{48} (x) \\ F_{59} (x) & = F_{12} (x) F_{47} (x) \\ F_{60} (x) & = F_{20} (x) + F_{61} (x) + F_{86} (x) \\ F_{61} (x) & = F_{12} (x) F_{62} (x) \\ F_{62} (x) & = F_{63} (x) + F_{68} (x) \\ F_{63} (x) & = F_{10} (x) + F_{64} (x) \\ F_{64} (x) & = F_{20} (x) + F_{65} (x) + F_{66} (x) \\ F_{65} (x) & = F_{12} (x) F_{63} (x) \\ F_{66} (x) & = F_{12} (x) F_{67} (x) \\ F_{67} (x) & = F_{45} (x) + F_{64} (x) \\ F_{68} (x) & = F_{69} (x) + F_{72} (x) \\ F_{69} (x) & = F_{20} (x) + F_{61} (x) + F_{70} (x) \\ F_{70} (x) & = F_{12} (x) F_{71} (x) \\ F_{71} (x) & = F_{41} (x) + F_{69} (x) \\ F_{72} (x) & = F_{20} (x) + F_{73} (x) + F_{83} (x) + F_{84} (x) \\ F_{73} (x) & = F_{12} (x) F_{74} (x) \\ F_{74} (x) & = F_{75} (x) + F_{78} (x) \\ F_{75} (x) & = F_{30} (x) + F_{76} (x) \\ F_{76} (x) & = F_{77} (x) \\ F_{77} (x) & = F_{12} (x) F_{64} (x) \\ F_{78} (x) & = F_{79} (x) + F_{81} (x) \\ F_{79} (x) & = 2 F_{20} (x) + F_{73} (x) + F_{80} (x) \\ F_{80} (x) & = F_{12} (x) F_{69} (x) \\ F_{81} (x) & = F_{82} (x) \\ F_{82} (x) & = F_{12} (x) F_{72} (x) \\ F_{83} (x) & = F_{12} (x) F_{68} (x) \\ F_{84} (x) & = F_{12} (x) F_{85} (x) \\ F_{85} (x) & = F_{48} (x) + F_{72} (x) \\ F_{86} (x) & = F_{12} (x) F_{87} (x) \\ F_{87} (x) & = F_{71} (x) + F_{88} (x) \\ F_{88} (x) & = F_{101} (x) + F_{89} (x) \\ F_{89} (x) & = F_{20} (x) + F_{49} (x) + F_{90} (x) \\ F_{90} (x) & = F_{12} (x) F_{91} (x) \\ F_{91} (x) & = F_{92} (x) + F_{93} (x) \\ F_{92} (x) & = F_{41} (x) + F_{55} (x) \\ F_{93} (x) & = F_{89} (x) + F_{94} (x) \\ F_{94} (x) & = F_{100} (x) + F_{20} (x) + F_{95} (x) + F_{96} (x) \\ F_{95} (x) & = 0 \\ F_{96} (x) & = F_{12} (x) F_{97} (x) \\ F_{97} (x) & = F_{98} (x) + F_{99} (x) \\ F_{98} (x) & = F_{55} (x) \\ F_{99} (x) & = F_{94} (x) \\ F_{100} (x) & = F_{12} (x) F_{89} (x) \\ F_{101} (x) & = F_{102} (x) + F_{113} (x) + F_{20} (x) + F_{73} (x) \\ F_{102} (x) & = F_{103} (x) F_{12} (x) \\ F_{103} (x) & = F_{104} (x) + F_{105} (x) \\ F_{104} (x) & = F_{69} (x) + F_{79} (x) \\ F_{105} (x) & = F_{101} (x) + F_{106} (x) \\ F_{106} (x) & = 2 F_{20} (x) + F_{107} (x) + F_{108} (x) + F_{112} (x) \\ F_{107} (x) & = 0 \\ F_{108} (x) & = F_{109} (x) F_{12} (x) \\ F_{109} (x) & = F_{110} (x) + F_{111} (x) \\ F_{110} (x) & = F_{79} (x) \\ F_{111} (x) & = F_{106} (x) \\ F_{112} (x) & = F_{101} (x) F_{12} (x) \\ F_{113} (x) & = F_{12} (x) F_{88} (x) \end{aligned}

Av(1243, 2143, 2341)

This specification was found using the strategy pack "Point Placements" and has 114 rules.

Proof Tree

System of Equations