

A180016


Partial sums of number of nstep closed paths on hexagonal lattice A002898.


0



1, 1, 7, 19, 109, 469, 2509, 12589, 67399, 358039, 1946395, 10622755, 58600531, 324978643, 1813780243, 10169519635, 57273912685, 323755931917, 1836345339961, 10446793409041, 59591722204861, 340755882430381
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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).
Recurrence: n^2*a(n) = (2*n1)*n*a(n1) + (n1)*(23*n24)*a(n2) + 12*(n4) * (n1)*a(n3)  36*(n2)*(n1)*a(n4).  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*x1)^2))/((1x)*(3*x+1)^(1/3)*(16*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)^(nni)*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: A026574 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



