A328268 Total number of nodes in all walks on cubic lattice starting at (0,0,0), ending at (0,0,n), remaining in the first (nonnegative) octant and using steps which are permutations of (-2,1,2), (-1,0,2), (-1,1,1), (0,0,1).

1, 2, 9, 36, 225, 1458, 11277, 92520, 829521, 7827750, 77868945, 805275756, 8628016761, 95164931610, 1076945117265, 12459200094864, 146980338550101, 1763951883200982, 21496281107991273, 265571585920327140, 3321596194293592593, 42009000779476030410

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..848

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Formula

a(n) = (n+1) * A328267(n).

Maple

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
      add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-2..2]))
    end:
a:= n-> (n+1)*b([0$2, n]):
seq(a(n), n=0..25);

Mathematica

b[l_] := b[l] = If[Last[l] == 0, 1, Function[r, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, r}, {j, r}, {k, r}]][Range[-2, 2]]];
a[n_] := (n + 1) b[{0, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, May 13 2020, after Maple *)

Crossrefs

Cf. A328267.

Sequence in context: A305206 A101610 A111601 * A330346 A280351 A306721

Adjacent sequences: A328265 A328266 A328267 * A328269 A328270 A328271

Keyword

nonn,walk

Author

Alois P. Heinz, Oct 10 2019

Status

approved