A149365 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, 0, 0), (-1, 0, 1), (0, 1, -1), (1, -1, 1), (1, 0, 1)}.

1, 1, 4, 12, 44, 171, 683, 2816, 11893, 51350, 224098, 991506, 4437757, 20006333, 90924488, 415980396, 1913682104, 8847746238, 41091054180, 191607760032, 896578541595, 4209023017990, 19817896294387, 93556842240115, 442750975652565, 2100004692492617, 9981048900376549, 47529264968202643

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..168

A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.

Maple

f:= proc(a, b, c, n) option remember;
   local t;
   if n = 0 then return 1 fi;
   t:= procname(a+1, b, c+1, n-1);
   if a >= 1 then t:= t + procname(a-1, b, c, n-1) + procname(a-1, b, c+1, n-1) fi;
   if c >= 1 then t:= t + procname(a, b+1, c-1, n-1) fi;
   if b >= 1 then t:= t + procname(a+1, b-1, c+1, n-1) fi;
   t;
end proc:
seq(f(0, 0, 0, n), n=0..50); # Robert Israel, Nov 07 2017

Mathematica

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[-1 + i, 1 + j, -1 + k, -1 + n] + aux[i, -1 + j, 1 + k, -1 + n] + aux[1 + i, j, -1 + k, -1 + n] + aux[1 + i, j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]

Crossrefs

Sequence in context: A151460 A149364 A055542 * A000759 A076793 A007860

Adjacent sequences: A149362 A149363 A149364 * A149366 A149367 A149368

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved