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%I #8 Dec 28 2023 19:44:52
%S 1,1,2,4,11,31,86,262,787,2442,7677,24581,80379,261809,867313,2890472,
%T 9684442,32683354,110898717,379064856,1297810439,4466393711,
%U 15433476163,53483388756,185962286216,648576840593,2269821822857,7957275602226,27969780462858,98546712605786,347890004162970,1230552458589243
%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, 0), (-1, 1, 1), (0, 1, -1), (1, 0, 0)}.
%H Robert Israel, <a href="/A148161/b148161.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, 0], [-1, 1, 1], [0, 1, -1], [1, 0, 0] ]:
%p f:= proc(n, p) option remember;
%p if n <= min(p) then return 4^n fi;
%p add(procname(n-1, t), t=remove(has, map(`+`, Steps, p), -1));
%p end proc:
%p map(f, [$0..30], [0, 0, 0]); # _Robert Israel_, Aug 22 2019
%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, j, k, -1 + n] + aux[i, -1 + j, 1 + k, -1 + n] + aux[1 + i, -1 + j, -1 + 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}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008
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