A150500 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, 1, 0), (1, 1, -1), (1, 1, 1)}.

1, 2, 7, 25, 101, 416, 1787, 7792, 34645, 155722, 707795, 3242515, 14963665, 69458000, 324102287, 1519028843, 7147771981, 33750528146, 159860887355, 759295147045, 3615520821281, 17255165910632, 82521746019487, 395404081034830, 1897886817388201, 9124229781131546, 43930513066698367, 211803668881914847

Offset

0,2

Links

Table of n, a(n) for n=0..27.

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

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Formula

a(n) = (A201805(n+1) + 3*A201805(n))/4. - Mark van Hoeij, Nov 29 2024

Conjecture D-finite with recurrence (n+1)^2*a(n) +(-9*n-4)*a(n-1) +(-22*n^2+27*n+15)*a(n-2) -3*(n-2)*(8*n-31)*a(n-3) +45*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Oct 29 2025

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

Cf. A201805.

Sequence in context: A150497 A150498 A150499 * A221455 A150501 A150502

Adjacent sequences: A150497 A150498 A150499 * A150501 A150502 A150503

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved