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

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

1, 2, 6, 23, 94, 391, 1674, 7340, 32810, 148460, 677221, 3113945, 14427443, 67248566, 314944257, 1480937061, 6990005401, 33106706997, 157268520653, 748999807143, 3575464723018, 17104963318797, 81991933706087, 393724722012390, 1893722027888591, 9122010044437447, 44001916585556279, 212527283465755898

0,2

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[-1 + i, j, -1 + k, -1 + n] + aux[i, -1 + j, k, -1 + n] + aux[1 + i, -1 + j, 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}]

nonn,walk

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

approved