A121822 Number of closed walks of length 2*n on the 5-cube.

1, 5, 65, 1205, 26465, 628805, 15424865, 382964405, 9550195265, 238539648005, 5961554097665, 149021418519605, 3725378557692065, 93133051794619205, 2328313585536338465, 58207725254446186805, 1455192101905494196865

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..700

G. R. Franssens, On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

L. Reyzin, Mathoverflow, Number of closed walks on an n-cube

Index entries for linear recurrences with constant coefficients, signature (35,-259,225).

Formula

a(n) = (25^n + 5*9^n + 10)/16.

G.f.: (1 - 30*x + 149*x^2)/(1 - 35*x + 259*x^2 - 225*x^3).

From Peter Bala, Nov 13 2006: (Start)

E.g.f.: cosh^5(x).

O.g.f.: 1/(1-5*1x/(1-4*2x/(1-3*3x/(1-2*4x/(1-1*5x))))) (continued fraction). (End)

(-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-6)^(n-k). - Philippe Deléham, Aug 17 2007

a(n) = (1/2^5)*Sum_{j = 0..5} binomial(5,j)*(5 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019

Mathematica

Table[(25^n +5*9^n +10)/16, {n, 0, 20}] (* G. C. Greubel, Jun 07 2019 *)

Prog

(PARI) a(n)=(25^n+5*9^n+10)>>4 \\ Charles R Greathouse IV, Jan 17 2012
(Magma) [(25^n +5*9^n +10)/16: n in [0..20]]; // G. C. Greubel, Jun 07 2019
(Sage) [(25^n +5*9^n +10)/16 for n in (0..20)] # G. C. Greubel, Jun 07 2019
(GAP) List([0..20], n-> (25^n +5*9^n +10)/16) # G. C. Greubel, Jun 07 2019

Crossrefs

Cf. A054879, A081294, A092812.

Sequence in context: A006278 A234871 A349517 * A056245 A357338 A195886

Adjacent sequences: A121819 A121820 A121821 * A121823 A121824 A121825

Keyword

nonn,easy

Author

Philippe Deléham, Aug 27 2006

Extensions

Corrected by T. D. Noe, Nov 07 2006

Status

approved