A091056 Expansion of x^2/((1-x)*(1+2*x)*(1-6*x)).

0, 0, 1, 5, 33, 193, 1169, 6993, 42001, 251921, 1511697, 9069841, 54419729, 326517009, 1959104785, 11754623249, 70527750417, 423166480657, 2538998927633, 15233993478417, 91403961045265, 548423765922065

Offset: 0

Author

Paul Barry, Dec 17 2003

Keywords

• easy
• nonn

Comments

4*A091055(n) counts walks of length n between non-adjacent vertices of the Johnson graph J(5,2).

6^n = A091054(n) + 6*A091055(n) + 12*a(n).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Johnson Graph
• Index entries for linear recurrences with constant coefficients, signature (5,8,-12).

Crossrefs

Cf. A091054, A091055.

Programs

GAP

List([0..40], n-> (3*6^n +5*(-2)^n -8)/120); # G. C. Greubel, Dec 27 2019

Magma

[(3*6^n +5*(-2)^n -8)/120: n in [0..40]]; // G. C. Greubel, Dec 27 2019

Maple

seq( (3*6^n +5*(-2)^n -8)/120, n=0..40); # G. C. Greubel, Dec 27 2019

Mathematica

CoefficientList[Series[x^2/((1-x)(1+2x)(1-6x)),{x,0,40}],x] (* or *) LinearRecurrence[{5,8,-12},{0,0,1},40] (* Harvey P. Dale, Apr 02 2015 *)

PARI

vector(41, n, (3*6^(n-1) + 5*(-2)^(n-1) - 8)/120) \\ G. C. Greubel, Dec 27 2019

Sage

[(3*6^n +5*(-2)^n -8)/120 for n in (0..40)] # G. C. Greubel, Dec 27 2019

Formula

a(n) = (3 * 6^n + 5*(-2)^n - 8)/120.

a(0)=0, a(1)=0, a(2)=1, a(n) = 5*a(n-1) + 8*a(n-2) - 12*a(n-3). - Harvey P. Dale, Apr 02 2015

E.g.f.: (3*exp(6*x) + 5*exp(-2*x) - 8*exp(x))/120. - G. C. Greubel, Dec 27 2019