PermPal Database

Av(1243, 2314, 4132)

Generating Function

- \frac{7 x^{6} - 25 x^{5} + 51 x^{4} - 56 x^{3} + 32 x^{2} - 9 x + 1}{(2 x - 1) {(x - 1)}^{2} {(x^{2} - 3 x + 1)}^{2}}

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Counting Sequence

1, 1, 2, 6, 21, 73, 244, 787, 2468, 7570, 22809, 67727, 198664, 576775, 1659914, ...

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Implicit Equation for the Generating Function

(2 x - 1) {(x - 1)}^{2} {(x^{2} - 3 x + 1)}^{2} F (x) + 7 x^{6} - 25 x^{5} + 51 x^{4} - 56 x^{3} + 32 x^{2} - 9 x + 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (4) = 21

a (5) = 73

a (6) = 244

a (n + 5) = 2 a (n) - 13 a (n + 1) + 28 a (n + 2) - 23 a (n + 3) + 8 a (n + 4) + n - 2, n \geq 7

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Explicit Closed Form

\frac{((25 n + 17) \sqrt{5} - 45 n - 75) {(\frac{3}{2} - \frac{\sqrt{5}}{2})}^{- n}}{50} + \frac{((- 25 n - 17) \sqrt{5} - 45 n - 75) {(\frac{3}{2} + \frac{\sqrt{5}}{2})}^{- n}}{50} - n + 2^{n} + 3

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This specification was found using the strategy pack "Point Placements" and has 69 rules.

Found on January 18, 2022.

Finding the specification took 3 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_{21} (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_{6} (x) \\ F_{14} (x) & = F_{15} (x) + F_{16} (x) + F_{20} (x) \\ F_{15} (x) & = 0 \\ F_{16} (x) & = F_{12} (x) F_{17} (x) \\ F_{17} (x) & = F_{18} (x) + F_{19} (x) \\ F_{18} (x) & = F_{10} (x) \\ F_{19} (x) & = F_{14} (x) \\ F_{20} (x) & = F_{12} (x) F_{13} (x) \\ F_{21} (x) & = F_{2} (x) + F_{22} (x) \\ F_{22} (x) & = F_{15} (x) + F_{23} (x) + F_{52} (x) \\ F_{23} (x) & = F_{12} (x) F_{24} (x) \\ F_{24} (x) & = F_{25} (x) + F_{32} (x) \\ F_{25} (x) & = F_{26} (x) + F_{6} (x) \\ F_{26} (x) & = F_{27} (x) \\ F_{27} (x) & = F_{12} (x) F_{28} (x) \\ F_{28} (x) & = F_{29} (x) \\ F_{29} (x) & = F_{30} (x) + F_{6} (x) \\ F_{30} (x) & = F_{31} (x) \\ F_{31} (x) & = F_{12} (x) F_{29} (x) \\ F_{32} (x) & = F_{22} (x) + F_{33} (x) \\ F_{33} (x) & = 2 F_{15} (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_{26} (x) \\ F_{37} (x) & = F_{33} (x) \\ F_{38} (x) & = F_{12} (x) F_{39} (x) \\ F_{39} (x) & = F_{40} (x) \\ F_{40} (x) & = F_{41} (x) + F_{46} (x) \\ F_{41} (x) & = F_{42} (x) \\ F_{42} (x) & = F_{12} (x) F_{43} (x) \\ F_{43} (x) & = F_{44} (x) + F_{45} (x) \\ F_{44} (x) & = F_{6} (x) \\ F_{45} (x) & = F_{41} (x) \\ F_{46} (x) & = 2 F_{15} (x) + F_{47} (x) + F_{51} (x) \\ F_{47} (x) & = F_{12} (x) F_{48} (x) \\ F_{48} (x) & = F_{49} (x) + F_{50} (x) \\ F_{49} (x) & = F_{30} (x) \\ F_{50} (x) & = F_{46} (x) \\ F_{51} (x) & = F_{12} (x) F_{40} (x) \\ F_{52} (x) & = F_{12} (x) F_{53} (x) \\ F_{53} (x) & = F_{54} (x) + F_{66} (x) \\ F_{54} (x) & = F_{2} (x) + F_{55} (x) \\ F_{55} (x) & = F_{15} (x) + F_{56} (x) + F_{65} (x) \\ F_{56} (x) & = F_{12} (x) F_{57} (x) \\ F_{57} (x) & = F_{58} (x) + F_{59} (x) \\ F_{58} (x) & = F_{10} (x) + F_{30} (x) \\ F_{59} (x) & = F_{55} (x) + F_{60} (x) \\ F_{60} (x) & = 2 F_{15} (x) + F_{51} (x) + F_{61} (x) \\ F_{61} (x) & = F_{12} (x) F_{62} (x) \\ F_{62} (x) & = F_{63} (x) + F_{64} (x) \\ F_{63} (x) & = F_{30} (x) \\ F_{64} (x) & = F_{60} (x) \\ F_{65} (x) & = F_{12} (x) F_{54} (x) \\ F_{66} (x) & = F_{41} (x) + F_{67} (x) \\ F_{67} (x) & = 2 F_{15} (x) + F_{47} (x) + F_{68} (x) \\ F_{68} (x) & = F_{12} (x) F_{66} (x) \end{aligned}

Av(1243, 2314, 4132)

This specification was found using the strategy pack "Point Placements" and has 69 rules.

Proof Tree

System of Equations