A148240 - OEIS

%I #9 Jan 01 2024 00:32:01

%S 1,1,2,4,13,36,110,367,1243,4266,14979,54636,199011,734030,2756839,

%T 10450319,39836796,153059496,593960500,2316916479,9077686257,

%U 35778018898,141845770750,564450779427,2253100544758,9036652220948,36388658308498,146848107101360,594357667427186,2414872472739607

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

%H Robert Israel, <a href="/A148240/b148240.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], 2008-2009.

%p Steps:= [[-1, -1, -1], [-1, -1, 1], [-1, 1, 1], [0, 1, -1], [1, 0, 0]]:

%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_, Apr 11 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, -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