A148647 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, 1), (-1, 0, 1), (-1, 1, 0), (1, 0, 0), (1, 1, -1)}.

1, 1, 3, 6, 23, 62, 263, 815, 3641, 12300, 56667, 202971, 954066, 3564277, 16985607, 65498160, 315209452, 1245818035, 6039009631, 24343527621, 118650810900, 486056238279, 2379084219687, 9877543772753, 48508370650463, 203692718352749, 1002991082498646, 4252712706392436, 20985613117596447

Offset

0,3

Links

Table of n, a(n) for n=0..28.

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, 1], [-1, 0, 1], [-1, 1, 0], [1, 0, 0], [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

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, j, k, -1 + n] + aux[1 + i, -1 + j, k, -1 + n] + aux[1 + i, j, -1 + k, -1 + n] + aux[1 + i, 1 + j, -1 + 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: A148644 A148645 A148646 * A148648 A013213 A013217

Adjacent sequences: A148644 A148645 A148646 * A148648 A148649 A148650

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved