A328295 Number of n-step walks on cubic lattice starting at (0,0,0), 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, 3, 12, 57, 294, 1590, 8856, 50301, 289590, 1683306, 9853320, 57977922, 342494118, 2029350972, 12052664868, 71715479535, 427347761010, 2549540104944, 15224944518084, 90988367614254, 544115710748898, 3255541325220204, 19486893225315138, 116685749052336714

Offset: 0

Author

Alois P. Heinz, Oct 11 2019

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..630
• Wikipedia, Lattice path
• Wikipedia, Self-avoiding walk

Crossrefs

Row sums of A328297.

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(add(b(sort([i, j, n-i-j])), j=0..n-i), i=0..n):
seq(a(n), n=0..23);

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}]][{-1, 0, 1}]];
a[n_] := Sum[Sum[b[Sort[{i, j, n - i - j}]], {j, 0, n - i}], {i, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, May 13 2020, after Maple *)