PermPal Database

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

Generating Function

\frac{x^{3} - 5 x^{2} + 4 x - 1}{(x - 1) {(2 x - 1)}^{2}}

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Counting Sequence

1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, ...

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Implicit Equation for the Generating Function

(x - 1) {(2 x - 1)}^{2} F (x) - x^{3} + 5 x^{2} - 4 x + 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 5

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

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Explicit Closed Form

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

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Found on January 18, 2022.

Finding the specification took 1 seconds.

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+ Expand

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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_{8} (x) \\ F_{4} (x) & = F_{5} (x) + F_{9} (x) \\ F_{5} (x) & = F_{1} (x) + F_{6} (x) \\ F_{6} (x) & = F_{7} (x) \\ F_{7} (x) & = F_{5} (x) F_{8} (x) \\ F_{8} (x) & = x \\ F_{9} (x) & = F_{10} (x) + F_{22} (x) \\ F_{10} (x) & = F_{11} (x) \\ F_{11} (x) & = F_{12} (x) F_{8} (x) \\ F_{12} (x) & = F_{13} (x) + F_{5} (x) \\ F_{13} (x) & = F_{14} (x) + F_{17} (x) \\ F_{14} (x) & = F_{15} (x) \\ F_{15} (x) & = F_{16} (x) F_{8} (x) \\ F_{16} (x) & = F_{1} (x) + F_{14} (x) \\ F_{17} (x) & = F_{18} (x) + F_{19} (x) + F_{21} (x) \\ F_{18} (x) & = 0 \\ F_{19} (x) & = F_{20} (x) F_{8} (x) \\ F_{20} (x) & = F_{17} (x) + F_{6} (x) \\ F_{21} (x) & = F_{13} (x) F_{8} (x) \\ F_{22} (x) & = F_{18} (x) + F_{23} (x) + F_{33} (x) \\ F_{23} (x) & = F_{24} (x) F_{8} (x) \\ F_{24} (x) & = F_{25} (x) + F_{28} (x) \\ F_{25} (x) & = F_{26} (x) + F_{6} (x) \\ F_{26} (x) & = F_{27} (x) \\ F_{27} (x) & = F_{25} (x) F_{8} (x) \\ F_{28} (x) & = F_{17} (x) + F_{29} (x) \\ F_{29} (x) & = 2 F_{18} (x) + F_{30} (x) + F_{32} (x) \\ F_{30} (x) & = F_{31} (x) F_{8} (x) \\ F_{31} (x) & = F_{26} (x) + F_{29} (x) \\ F_{32} (x) & = F_{28} (x) F_{8} (x) \\ F_{33} (x) & = F_{8} (x) F_{9} (x) \end{aligned}

Proof Tree