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

1, 1, 3, 8, 26, 78, 271, 896, 3228, 11119, 41002, 145548, 545187, 1974923, 7480699, 27503349, 105050951, 390597492, 1501505042, 5632087999, 21759951598, 82196014521, 318859845293, 1211380162124, 4714855246296, 17997471462958, 70241280052488, 269196555472317, 1053057343103259, 4049528715390724

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