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%I #12 Dec 13 2018 17:35:14
%S 1,1,2,4,12,35,113,375,1310,4703,17330,65209,250114,973787,3843412,
%T 15348830,61943321,252282099,1036030465,4286325796,17852945250,
%U 74812425886,315247690475,1335177040541,5681387342453,24279598309480,104174406713499,448629070128615,1938691951621376,8404757306420331
%N 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, 0, 0), (-1, 0, 1), (0, 0, 1), (0, 1, -1), (1, -1, 1)}.
%H Robert Israel, <a href="/A148204/b148204.txt">Table of n, a(n) for n = 0..200</a>
%H Alin Bostan and Manuel Kauers, <a href="https://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2009.
%p Steps:= [[-1, 0, 0], [-1, 0, 1], [0, 0, 1], [0, 1, -1], [1, -1, 1]]:
%p f:= proc(n, p) option remember;
%p if n <= min(p) then return 5^n fi;
%p add(procname(n-1, t), t=remove(has, map(`+`, Steps, p), -1)); end proc:
%p map(f, [$0..40], [0, 0, 0]); # _Robert Israel_, Dec 13 2018
%t 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, -1 + j, 1 + k, -1 + n] + aux[i, j, -1 + k, -1 + n] + aux[1 + i, j, -1 + k, -1 + n] + aux[1 + i, j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008