editing

approved

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

approved

editing

_Manuel Kauers (manuel(AT)kauers.de), _, Nov 18 2008

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, 1), (0, 0, 1), (1, 1, -1)}

1, 1, 2, 6, 18, 55, 191, 641, 2338, 8412, 31626, 118683, 457359, 1763742, 6931137, 27261381, 108770025, 434387917, 1753998833, 7090521404, 28908852160, 118022306253, 485066186991, 1996484012579, 8261227549240, 34234587581788, 142482320374052, 593868063996294, 2484063713185947

0,3

A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, <a href="http://arxiv.org/abs/0811.2899">ArXiv 0811.2899</a>.

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[1 + i, -1 + j, -1 + 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}]

nonn,walk

Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008

approved