A148162 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, 1, 1), (0, 0, -1), (1, 0, 0)}.

1, 1, 2, 4, 11, 31, 91, 267, 813, 2557, 8236, 26786, 87847, 291227, 977068, 3308866, 11278661, 38653717, 133227658, 461825572, 1608935543, 5628584683, 19762998641, 69637415101, 246214217449, 873256171561, 3105884952236, 11074941003322, 39586724293339, 141826521919151, 509214076570561, 1831934157570821

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..220

Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.

Formula

Empirical: -63*(k+3)*(k+2)*(k+1)*a(k)-2*(2*k-1)*(k+3)*(k+2)*a(k+1)+(k+3)*(6*k^2+32*k+31)*a(k+2)-(2*k+9)*(2*k^2+18*k+39)*a(k+3)+(k+4)*(k+6)^2*a(k+4) = 0. - Robert Israel, Apr 11 2019

Maple

Steps:= [[-1, -1, 1], [-1, 1, 1], [0, 0, -1], [1, 0, 0]]:
f:= proc(n, p) option remember;
  if n <= min(p) then return 4^n fi;
  add(procname(n-1, t), t=remove(has, map(`+`, Steps, p), -1));
end proc:
A:= map(f, [$0..100], [0, 0, 0]); # Robert Israel, Apr 11 2019

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, k, -1 + n] + aux[i, j, 1 + k, -1 + n] + aux[1 + i, -1 + 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}]

Crossrefs

Sequence in context: A148160 A148161 A263375 * A148163 A274775 A039300

Adjacent sequences: A148159 A148160 A148161 * A148163 A148164 A148165

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved