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
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.
Programs
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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 *)
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
Extensions
More terms and formula from Henry Bottomley, Aug 23 2001