A052177 Number of walks on simple cubic lattice (starting on the xy plane, never going below it and finishing a height 1 above it).

0, 1, 8, 50, 288, 1605, 8824, 48286, 264128, 1447338, 7953040, 43842788, 242507456, 1345868589, 7493458392, 41850173670, 234408444288, 1316541032958, 7413214297968, 41842633282620, 236703844320960

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Formula

a(n) = 4*a(n-1)+A005572(n-1)+A052178(n-1) = A052179(n, 1) = Sum_{j=0..ceiling((n-1)/2)} 4^(n-2j-1)*binomial(n, 2j+1)*binomial(2j+2, j+1)/(j+2).

Recurrence: (n-1)*(n+3)*a(n) = 4*n*(2*n+1)*a(n-1) - 12*(n-1)*n*a(n-2). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ 6^(n+3/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012

G.f.: (1 - 4*x - sqrt(1-8*x+12*x^2))^2/(4*x^3). - Mark van Hoeij, May 16 2013

Mathematica

Flatten[{0, RecurrenceTable[{(n-1)*(n+3)*a[n] == 4*n*(2*n+1)*a[n-1] - 12*(n-1)*n*a[n-2], a[1]==1, a[2]==8}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)

Crossrefs

Sequence in context: A081675 A283277 A081180 * A127745 A243876 A037547

Adjacent sequences: A052174 A052175 A052176 * A052178 A052179 A052180

Keyword

nonn,walk

Author

N. J. A. Sloane, Jan 26 2000

Extensions

More terms and formula from Henry Bottomley, Aug 23 2001

Status

approved