A151367 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, 1)}.

1, 0, 2, 2, 13, 27, 140, 392, 1882, 6289, 28906, 107949, 486438, 1948638, 8730438, 36611160, 164259758, 710530289, 3203433595, 14163150429, 64260242637, 288694503092, 1318679597635, 5996837692998, 27572301084897, 126595556379751, 585652882733959, 2709967750078764, 12607711205847168

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..700

A. Bostan, K. Raschel, and B. Salvy, Non-D-finite excursions in the quarter plane, J. Comb. Theory A 121 (2014) 45-63, Table 1 Tag 43, Tag 49.

Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Maple

Steps:= [[-1, -1], [-1, 0], [-1, 1], [0, -1], [0, 1], [1, 1]]:
f:= proc(n, p) option remember; local t, s;
      if max(p) > n then return 0 fi;
      add(procname(n-1, s), s = select(t -> min(t)>=0, map(`+`, Steps, p)))
end proc:
f(0, [0, 0]):= 1:
map(f, [$0..40], [0, 0]); # Robert Israel, Aug 02 2024

Mathematica

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]

Crossrefs

Sequence in context: A151352 A155915 A173466 * A368957 A057648 A282460

Adjacent sequences: A151364 A151365 A151366 * A151368 A151369 A151370

Keyword

nonn,walk,changed

Author

Manuel Kauers, Nov 18 2008

Status

approved