A092803 Expansion of (1-5*x)/((1-2*x)*(1-6*x)).

1, 3, 12, 60, 336, 1968, 11712, 70080, 420096, 2519808, 15117312, 90700800, 544198656, 3265179648, 19591053312, 117546270720, 705277526016, 4231664959488, 25389989363712, 152339935395840, 914039610802176

Offset: 0

Author

Paul Barry, Mar 06 2004

Keywords

• easy
• nonn

Comments

Second binomial transform of expansion of (1-3*x)/(1-4*x). Third binomial transform of A054878 (closed walks at a vertex of K_4). With interpolated zeros, counts closed walks of length n at the vertices of the edge-vertex incidence graph of K_4 associated with the vertices of K_4.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-12).

Crossrefs

Cf. A054878, A092807.

Programs

Magma

[3*(6^(n-1) + 2^(n-1))/2: n in [0..40]]; // G. C. Greubel, Jan 04 2023

Mathematica

LinearRecurrence[{8,-12}, {1,3}, 41] (* G. C. Greubel, Jan 04 2023 *)
CoefficientList[Series[(1-5x)/((1-2x)(1-6x)),{x,0,30}],x] (* Harvey P. Dale, Jan 30 2024 *)

SageMath

[3*(6^(n-1) +2^(n-1))/2 for n in range(41)] # G. C. Greubel, Jan 04 2023

Formula

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

G.f.: U(0)/4 where U(k)= 1 + 2/(3^k - 3^k/(2 + 1 - 12*x*3^k/(6*x*3^k + 1/U(k+1)))) ; (continued fraction, 4-step). - Sergei N. Gladkovskii, Oct 30 2012

G.f.: U(0)/4 where U(k)= 1 + 3/( 3^k - 2*x*9^k/(2*x*3^k + 1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 31 2012

E.g.f.: (1/4)*( 3*exp(2*x) + exp(6*x) ). - G. C. Greubel, Jan 04 2023