A328281 Total number of nodes in all 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, 2, 9, 28, 130, 492, 2401, 10056, 50346, 224110, 1141030, 5284200, 27222364, 129582278, 673340265, 3270171504, 17104148290, 84342959172, 443406172278, 2212675109180, 11680186909062, 58844537998598, 311667574680190, 1582458820299408, 8404755004516300

Offset: 0

Author

Alois P. Heinz, Oct 10 2019

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Lattice path
• Wikipedia, Self-avoiding walk

Crossrefs

Cf. A328280.

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

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_] := Function[t, (n + 1) b[{0, t, n - t}]][Quotient[n, 2]];
a /@ Range[0, 31] (* Jean-François Alcover, May 13 2020, after Maple *)

Formula

a(n) = (n+1) * A328280(n).