A148370 - OEIS

%I #7 Jan 01 2024 00:49:34

%S 1,1,2,5,16,42,147,454,1628,5425,20178,70153,268235,963094,3750168,

%T 13820410,54545612,205188861,818752204,3130596415,12607433542,

%U 48850200688,198265158766,776800033465,3173598172091,12552548689917,51573263660315,205664370031343,849144348277944,3410421662866518

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

%H Robert Israel, <a href="/A148370/b148370.txt">Table of n, a(n) for n = 0..210</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, 0, 1], [-1, 1, -1], [0, 1, 1], [1, 0, -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_, Jul 11 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, j, 1 + k, -1 + n] + aux[i, -1 + j, -1 + k, -1 + n] + aux[1 + i, -1 + j, 1 + k, -1 + n] + aux[1 + i, 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