A151270 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, 0), (0, -1), (1, 1)}.

1, 1, 4, 9, 30, 90, 286, 941, 3072, 10308, 34814, 118138, 406368, 1399630, 4858636, 16948153, 59302432, 208587930, 735602234, 2602943312, 9239044912, 32867768218, 117244508016, 419105764722, 1501220073148, 5388115272948, 19371438711296, 69764776854918, 251641664047600, 908984140277160, 3288068461179108

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..500

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. 7 (Theorem 7, Table 4).

Maple

steps:= [[-1, -1], [-1, 0], [0, -1], [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, Jun 03 2026

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[1 + i, 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: A295910 A086688 A309295 * A149113 A149114 A105865

Adjacent sequences: A151267 A151268 A151269 * A151271 A151272 A151273

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved