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

0,3

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.

Links

Table of n, a(n) for n=0..21.

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

Example

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.

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 *)

Crossrefs

Cf. A002898, A115130, A174459, A174655.

Sequence in context: A351858 A240150 A091149 * A180025 A070976 A249608

Adjacent sequences: A180013 A180014 A180015 * A180017 A180018 A180019

Keyword

nonn,walk

Author

Jonathan Vos Post, Jan 13 2011

Status

approved