A151289 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 1), (1, 0), (1, 1)}.

1, 2, 7, 21, 80, 267, 1042, 3655, 14400, 51960, 205770, 756169, 3003754, 11179972, 44496878, 167181919, 666257284, 2521360818, 10057256094, 38278262216, 152783755958, 584199222356, 2332875408218, 8954456162165, 35770374449310, 137743370071244, 550391555936678, 2125263063052980

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..800

Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.

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

Marni Mishna and Juan Pulido, On the small-step quarter plane lattice walks with a non D-finite univariate generating function, arXiv:2605.16688 [math.CO], 2026. See p. 14 (Table 6).

Maple

steps:= [[-1, -1], [-1, 1], [1, 0], [1, 1]]:
f:= proc(n, p) option remember; local t;
  if n <= min(p) then return 4^n fi;
  add(procname(n-1, t), t=remove(has, map(`+`, steps, p), -1));
end proc:
map(f, [$0..100], [0, 0]); # Robert Israel, Nov 09 2025

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[-1 + i, j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A319852 A060900 A305850 * A150300 A150301 A150302

Adjacent sequences: A151286 A151287 A151288 * A151290 A151291 A151292

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved