PermPal Database

Av(1234, 1324, 2413)

Generating Function

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

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Counting Sequence

1, 1, 2, 6, 21, 75, 265, 925, 3201, 11017, 37793, 129393, 442497, 1512225, 5165953, ...

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Implicit Equation for the Generating Function

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

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (4) = 21

a (n + 3) = 4 a (n) - 10 a (n + 1) + 6 a (n + 2) + 1, n \geq 5

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Explicit Closed Form

{\begin{array}{cc} 1 & n = 0 \\ 1 - \frac{{(1 + \frac{\sqrt{2}}{2})}^{- n} \sqrt{2}}{8} + \frac{{(1 - \frac{\sqrt{2}}{2})}^{- n} \sqrt{2}}{8} - \frac{2^{n}}{4} & otherwise \end{array}

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This specification was found using the strategy pack "Point Placements" and has 51 rules.

Found on January 18, 2022.

Finding the specification took 5 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_{16} (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_{15} (x) & = F_{12} (x) F_{13} (x) \\ F_{16} (x) & = F_{17} (x) + F_{2} (x) \\ F_{17} (x) & = F_{18} (x) + F_{19} (x) + F_{46} (x) \\ F_{18} (x) & = 0 \\ F_{19} (x) & = F_{12} (x) F_{20} (x) \\ F_{20} (x) & = F_{21} (x) + F_{25} (x) \\ F_{21} (x) & = F_{22} (x) + F_{6} (x) \\ F_{22} (x) & = F_{23} (x) \\ F_{23} (x) & = F_{12} (x) F_{24} (x) \\ F_{24} (x) & = F_{10} (x) \\ F_{25} (x) & = F_{17} (x) + F_{26} (x) \\ F_{26} (x) & = 2 F_{18} (x) + F_{27} (x) + F_{31} (x) \\ F_{27} (x) & = F_{12} (x) F_{28} (x) \\ F_{28} (x) & = F_{29} (x) + F_{30} (x) \\ F_{29} (x) & = F_{22} (x) \\ F_{30} (x) & = F_{26} (x) \\ F_{31} (x) & = F_{12} (x) F_{32} (x) \\ F_{32} (x) & = F_{33} (x) \\ F_{33} (x) & = F_{18} (x) + F_{34} (x) + F_{44} (x) \\ F_{34} (x) & = F_{12} (x) F_{35} (x) \\ F_{35} (x) & = F_{36} (x) + F_{38} (x) \\ F_{36} (x) & = F_{10} (x) + F_{37} (x) \\ F_{37} (x) & = F_{23} (x) \\ F_{38} (x) & = F_{33} (x) + F_{39} (x) \\ F_{39} (x) & = 2 F_{18} (x) + F_{31} (x) + F_{40} (x) \\ F_{40} (x) & = F_{12} (x) F_{41} (x) \\ F_{41} (x) & = F_{42} (x) + F_{43} (x) \\ F_{42} (x) & = F_{37} (x) \\ F_{43} (x) & = F_{39} (x) \\ F_{44} (x) & = F_{12} (x) F_{45} (x) \\ F_{45} (x) & = F_{2} (x) + F_{33} (x) \\ F_{46} (x) & = F_{12} (x) F_{47} (x) \\ F_{47} (x) & = F_{45} (x) + F_{48} (x) \\ F_{48} (x) & = F_{17} (x) + F_{49} (x) \\ F_{49} (x) & = F_{50} (x) \\ F_{50} (x) & = F_{12} (x) F_{48} (x) \end{aligned}

Av(1234, 1324, 2413)

This specification was found using the strategy pack "Point Placements" and has 51 rules.

Proof Tree

System of Equations