A149595 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, -1, 0), (-1, 0, -1), (0, -1, 0), (1, 1, -1), (1, 1, 1)}.

1, 1, 5, 15, 59, 219, 921, 3729, 15511, 66307, 285723, 1231425, 5377117, 23746851, 104998531, 465502605, 2084418377, 9360389879, 42066742411, 189986019637, 861969546027, 3914101686361, 17807890803795, 81343463279557, 372217083213775, 1704575842549479, 7825997996053383, 36010679557986875

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.

Maple

N:= 30: # to get a(0) to a(N)
steps:= [[-1, -1, 0], [-1, 0, -1], [0, -1, 0], [1, 1, -1], [1, 1, 1]]:
P[0]:= {[0, 0, 0]}:
A[0]:= 1:
B[0, [0, 0, 0]]:= 1:
for n from 1 to N do
  A[n]:= 0:
  P[n]:= {}:
  for p in P[n-1] do
    for s in steps do
      pp:= p + s;
      if min(pp) < 0 then next fi;
      P[n]:= P[n] union {pp};
      A[n]:= A[n] + B[n-1, p];
      if assigned(B[n, pp]) then B[n, pp]:= B[n, pp] + B[n-1, p]
      else B[n, pp]:= B[n-1, p]
      fi;
    od
  od
od:
seq(A[n], n=0..N); # Robert Israel, Nov 03 2014
# Alternative:
b:= proc(n, l) option remember; `if`(n=0, 1, add((p->
      `if`(min(p[])<0, 0, b(n-1, p)))(l+h), h=[[-1$2, 0],
       [-1, 0, -1], [0, -1, 0], [1$2, -1], [1$3]]))
    end:
a:= n-> b(n, [0$3]):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2014

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[-1 + i, -1 + j, 1 + k, -1 + n] + aux[i, 1 + j, k, -1 + n] + aux[1 + i, j, 1 + k, -1 + n] + aux[1 + i, 1 + j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]

Crossrefs

Sequence in context: A149593 A230986 A149594 * A149596 A149597 A149598

Adjacent sequences: A149592 A149593 A149594 * A149596 A149597 A149598

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved