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%I #8 Feb 28 2022 02:06:22
%S 1,1,5,15,55,203,791,3041,12057,48249,194705,790373,3237213,13323957,
%T 55078721,228709819,953594747,3988265595,16730145283,70382381531,
%U 296830291631,1254604475451,5314036803363,22552374316543,95878895567139,408287637231539,1741343710323743,7437497858831271
%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), (-1, 0, 1), (0, 0, -1), (1, 1, 1)}
%H Robert Israel, <a href="/A149585/b149585.txt">Table of n, a(n) for n = 0..225</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, -1, 0], [-1, -1, 1], [-1, 0, 1], [0, 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..30], [0, 0, 0]); # _Robert Israel_, Feb 27 2022
%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, j, 1 + k, -1 + n] + aux[1 + i, 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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