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%I #7 Mar 29 2024 19:56:47
%S 1,1,3,8,28,91,339,1191,4673,17489,70829,275583,1140249,4555059,
%T 19176287,78180708,333655636,1382789189,5964790733,25047746365,
%U 108998800785,462656900979,2028379249377,8687945066980,38332181892050,165464671038519,734031617783073,3189844310837718,14217991701913836
%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, -1, -1), (-1, -1, 1), (-1, 0, 0), (1, 0, 1), (1, 1, -1)}.
%H Robert Israel, <a href="/A148850/b148850.txt">Table of n, a(n) for n = 0..220</a>
%H A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, <a href="http://arxiv.org/abs/0811.2899">ArXiv 0811.2899</a>.
%p Steps:= [[-1, -1, -1], [-1, -1, 1], [-1, 0, 0], [1, 0, 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));
%p end proc:
%p map(f, [$0..40], [0, 0, 0]); # _Robert Israel_, May 02 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[-1 + i, j, -1 + k, -1 + n] + aux[1 + i, j, k, -1 + n] + aux[1 + i, 1 + 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}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008