A238115 Number of states arising in matrix method for enumerating Hamiltonian cycles on a 2n X 2n grid.
1, 6, 32, 182, 1117, 7280, 49625, 349998, 2535077, 18758264, 141254654, 1079364104, 8350678169, 65298467486, 515349097712, 4100346740510, 32858696386765, 265001681344568, 2149447880547398, 17524254766905368, 143540915998174577, 1180736721910617182
Offset: 1
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..500
- Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv preprint arXiv:1402.0545 [math.CO], 2014.
Programs
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Maple
a := n -> hypergeom([1/2, -n, -n], [1, 2], 4) - 1: seq(simplify(a(n)), n = 1..22); # Peter Luschny, Dec 13 2024
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PARI
a(n)=sum(k=1,n,binomial(n,k)^2*binomial(2*k,k)/(k+1)) \\ Andrew Howroyd, Dec 13 2024
Formula
From Andrew Howroyd, Dec 13 2024: (Start)
a(n) = Sum_{k=1..n} binomial(n,k)^2 * A000108(k).
a(n) = A086618(n) - 1. (End)