A328296 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (0,y,z), 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, 2, 5, 16, 58, 228, 945, 4072, 18078, 82172, 380666, 1791138, 8537912, 41146988, 200169891, 981705400, 4848820372, 24098703860, 120433164750, 604831645542, 3050979757728, 15451575335362, 78536766518038, 400497435480332, 2048473941706016, 10506489209380466

Offset: 0

Author

Alois P. Heinz, Oct 11 2019

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..610
• Wikipedia, Lattice path
• Wikipedia, Self-avoiding walk

Crossrefs

Column k=0 of A328297.

Row sums of A328300.

Cf. A000079, A000918.

Programs

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-> add(b(sort([0, j, n-j])), j=0..n):
seq(a(n), n=0..29);

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_] := Sum[b[Sort[{0, j, n  - j}]], {j, 0, n }];
a /@ Range[0, 29] (* Jean-François Alcover, May 12 2020, after Maple *)

Formula

a(n) is odd <=> n in { A000918 } and n >= 0.