This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A180016 #20 Sep 27 2020 15:29:27
%S A180016 1,1,7,19,109,469,2509,12589,67399,358039,1946395,10622755,58600531,
%T A180016 324978643,1813780243,10169519635,57273912685,323755931917,
%U A180016 1836345339961,10446793409041,59591722204861,340755882430381
%N A180016 Partial sums of number of n-step closed paths on hexagonal lattice A002898.
%C A180016 Also, number of closed paths of length <= n on the honeycomb lattice. The analog on the square lattice is A115130.
%C A180016 The subsequence of primes begins 7, 19, 109, 12589, 67399.
%F A180016 a(n) = Sum_{i=0..n} A002898(i).
%F A180016 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
%F A180016 a(n) ~ 3*sqrt(3)*6^n/(5*Pi*n). - _Vaclav Kotesovec_, Oct 24 2012
%F A180016 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
%e A180016 a(0) = 1 because there is a unique null walk on no points.
%e A180016 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).
%e A180016 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).
%e A180016 a(8) = 1 + 0 + 6 + 12 + 90 + 360 + 2040 + 10080 + 54810 = 67399.
%t A180016 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 *)
%Y A180016 Cf. A002898, A115130, A174459, A174655.
%K A180016 nonn,walk
%O A180016 0,3
%A A180016 _Jonathan Vos Post_, Jan 13 2011