A328280 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (0,floor(n/2),ceiling(n/2)), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

1, 1, 3, 7, 26, 82, 343, 1257, 5594, 22411, 103730, 440350, 2094028, 9255877, 44889351, 204385719, 1006126370, 4685719954, 23337166962, 110633755459, 556199376622, 2674751727209, 13550764116530, 65935784179142, 336190200180652, 1651985253047884

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Formula

a(n) = A328300(n,floor(n/2)).

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))([$-1..1]))
    end:
a:= n-> (t-> b([0, t, n-t]))(iquo(n, 2)):
seq(a(n), n=0..31);

Mathematica

b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
a[n_] := With[{t = Quotient[n, 2]}, b[{0, t, n - t}]];
a /@ Range[0, 31] (* Jean-François Alcover, May 12 2020, after Maple *)

Crossrefs

Bisection gives A328269 (even part).

Cf. A328281, A328300.

Sequence in context: A208979 A148740 A148741 * A258311 A184459 A215018

Adjacent sequences: A328277 A328278 A328279 * A328281 A328282 A328283

Keyword

nonn,walk

Author

Alois P. Heinz, Oct 10 2019

Status

approved