A149424 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (0, -1, 0), (0, 0, -1), (1, 1, 1)}.

1, 1, 4, 13, 40, 136, 496, 1753, 6256, 22912, 85216, 314836, 1170688, 4396048, 16623328, 62744017, 237680992, 904962400, 3459831424, 13219219972, 50621972224, 194465172304, 749061374848, 2884682636764, 11126422372864, 43007603099296, 166555051934848, 644984620465264, 2500560314630656

Offset: 0

Author

Manuel Kauers, Nov 18 2008

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.

Crossrefs

Cf. A151265.

Column k=3 of A335570.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, map(x-> x+1, l))+
      add(`if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..3))
    end:
a:= n-> b(n, [0$3]):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2021

Mathematica

aux[i_Integer, j_Integer, k_Integer, n_Integer] := Which[Min[i, j, k, n] < 0 || Max[i, j, k] > n, 0, n == 0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1 + i, -1 + j, -1 + k, -1 + n] + aux[i, j, 1 + k, -1 + n] + aux[i, 1 + j, k, -1 + n] + aux[1 + i, j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]

Formula

a(n) == 1 (mod 3). - Alois P. Heinz, Jul 12 2021