A180016 Partial sums of number of n-step closed paths on hexagonal lattice A002898.

1, 1, 7, 19, 109, 469, 2509, 12589, 67399, 358039, 1946395, 10622755, 58600531, 324978643, 1813780243, 10169519635, 57273912685, 323755931917, 1836345339961, 10446793409041, 59591722204861, 340755882430381

Offset: 0

Author

Jonathan Vos Post, Jan 13 2011

Keywords

• nonn
• walk

Comments

Also, number of closed paths of length <= n on the honeycomb lattice. The analog on the square lattice is A115130.

The subsequence of primes begins 7, 19, 109, 12589, 67399.

Examples

			a(0) = 1 because there is a unique null walk on no points.
a(1) = 1 because there are no closed paths of length 1 (which connects the origin with one of 6 other points before symmetry is considered).
a(2) = 7 because one adds the 6 closed paths of length 2 (which go from origin to one of 6 surrounding points on the lattice, and return in the opposite directions).
a(8) = 1 + 0 + 6 + 12 + 90 + 360 + 2040 + 10080 + 54810 = 67399.
		

Crossrefs

Cf. A002898, A115130, A174459, A174655.

Programs

Mathematica

Table[Sum[Sum[(-2)^(nn-i)*Binomial[i, j]^3*Binomial[nn, i], {i, 0, nn}, {j, 0, i}],{nn,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n) = Sum_{i=0..n} A002898(i).

D-finite with recurrence: n^2*a(n) = (2*n-1)*n*a(n-1) + (n-1)*(23*n-24)*a(n-2) + 12*(n-4) * (n-1)*a(n-3) - 36*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ 3*sqrt(3)*6^n/(5*Pi*n). - Vaclav Kotesovec, Oct 24 2012

G.f.: hypergeom([1/3,1/3],[1],-27*x*(2*x+1)^2/((3*x+1)*(6*x-1)^2))/((1-x)*(3*x+1)^(1/3)*(1-6*x)^(2/3)). - Mark van Hoeij, Apr 17 2013