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

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

1, 2, 7, 22, 87, 323, 1370, 5539, 24459, 103964, 470998, 2068727, 9539561, 42876377, 200280183, 915568425, 4318502566, 19998346530, 95042265552, 444597529434, 2125686876422, 10024484835758, 48162780320992, 228632062310517, 1102892132039964, 5264147613227849, 25479145954143790, 122170625962148962

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