A094060 Number of walks of length n on hexagonal grid that start and end at the origin. Intermediate returns to the origin are not permitted.

1, 0, 6, 12, 54, 216, 1032, 4896, 24606, 125040, 651348, 3432168, 18331992, 98814816, 537343632, 2942475552, 16214888286, 89835783264, 500116783740, 2795958732024, 15690597591636, 88354191756816, 499060719941616, 2826794871554112, 16052536475622792

Offset: 0

Author

Gareth McCaughan, Jun 10 2004

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1291 (first 97 terms from Ludovic Schwob)

Crossrefs

Cf. A002898, A054474.

Programs

Maple

b:= proc(n) option remember; `if`(n<3, [1,0,6][n+1], ((n-1)*
      n*b(n-1) +24*(n-1)^2*b(n-2) +36*(n-1)*(n-2)*b(n-3))/n^2)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      b(n)-add(a(n-i)*b(i), i=1..n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Dec 07 2020

Mathematica

b[n_] := b[n] = If[n<3, {1, 0, 6}[[n+1]], ((n-1)n b[n-1] + 24(n-1)^2* b[n-2] + 36(n-1)(n-2) b[n-3])/n^2];
a[n_] := a[n] = If[n==0, 1, b[n] - Sum[a[n-i] b[i], {i, 1, n-1}]];
a /@ Range[0, 23] (* Jean-François Alcover, Jan 14 2021, after Alois P. Heinz *)

PARI

seq(n)={my(g=sum(m=0, n, (3*m)!/m!^3*x^(2*m)*(1+2*x)^m, O(x*x^n))); Vec(2-1/g)} \\ Andrew Howroyd, Aug 09 2025

Formula

INVERTi transform of A002898. - R. J. Mathar, Sep 29 2020