A238115 - OEIS

%I #20 Dec 13 2024 12:39:11

%S 1,6,32,182,1117,7280,49625,349998,2535077,18758264,141254654,

%T 1079364104,8350678169,65298467486,515349097712,4100346740510,

%U 32858696386765,265001681344568,2149447880547398,17524254766905368,143540915998174577,1180736721910617182

%N Number of states arising in matrix method for enumerating Hamiltonian cycles on a 2n X 2n grid.

%H Andrew Howroyd, <a href="/A238115/b238115.txt">Table of n, a(n) for n = 1..500</a>

%H Ed Wynn, <a href="http://arxiv.org/abs/1402.0545">Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs</a>, arXiv preprint arXiv:1402.0545 [math.CO], 2014.

%F From _Andrew Howroyd_, Dec 13 2024: (Start)

%F a(n) = Sum_{k=1..n} binomial(n,k)^2 * A000108(k).

%F a(n) = A086618(n) - 1. (End)

%p a := n -> hypergeom([1/2, -n, -n], [1, 2], 4) - 1:

%p seq(simplify(a(n)), n = 1..22);  # _Peter Luschny_, Dec 13 2024

%o (PARI) a(n)=sum(k=1,n,binomial(n,k)^2*binomial(2*k,k)/(k+1)) \\ _Andrew Howroyd_, Dec 13 2024

%Y Cf. A000108, A003763, A086618, A143246, A209077, A222065, A238116, A238117, A238118.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Mar 05 2014