A064046 Number of length 6 walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.

0, 5, 70, 285, 740, 1525, 2730, 4445, 6760, 9765, 13550, 18205, 23820, 30485, 38290, 47325, 57680, 69445, 82710, 97565, 114100, 132405, 152570, 174685, 198840, 225125, 253630, 284445, 317660, 353365, 391650, 432605, 476320, 522885, 572390

Offset: 0

Author

Henry Bottomley, Aug 23 2001

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Numbers of walks of length 0, 1, 2, 3, 4 and 5 are A000012, A000004, A001477, A000004, A049450 and A000004.

Programs

Magma

[5*n*(3*n^2-3*n+1): n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

Mathematica

LinearRecurrence[{4,-6,4,-1},{0,5,70,285},40] (* Harvey P. Dale, Dec 02 2012 *)

Formula

a(n) = 5*n*(3*n^2 - 3*n + 1).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = A064045(n, 3).

a(n) = a(n-1) + 15*A049450(n-1) + 30*A001477(n-1) + 5*A000012(n-1).

G.f.: 5*x*(7*x^2 + 10*x + 1)/(x-1)^4. [Colin Barker, Jul 21 2012]