A100303 Expansion of (1 - x - 4*x^2)/(1 - x - 8*x^2).

1, 0, 4, 4, 36, 68, 356, 900, 3748, 10948, 40932, 128516, 455972, 1484100, 5131876, 17004676, 58059684, 194097092, 658574564, 2211351300, 7479947812, 25170758212, 85010340708, 286376406404, 966459132068, 3257470383300, 10989143439844, 37048906506244, 124962054024996

Offset: 0

Author

Paul Barry, Nov 12 2004

Keywords

• easy
• nonn

Comments

Construct a graph as follows: form the graph whose adjacency matrix is the tensor product of that of P_3 and [1,1;1,1], then add a loop at each of the extremity nodes. (Spectrum : [0; 1^3; (1-sqrt(33))/2;(1+sqrt(33))/2]). a(n) counts closed walks of length n at each of the internal nodes.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,8).

Crossrefs

Cf. A015443, A100302, A100304.

Programs

Magma

[1] cat [n le 2 select 4*(n-1) else Self(n-1) +8*Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 04 2023

Mathematica

CoefficientList[Series[(1-x-4x^2)/(1-x-8x^2),{x,0,30}],x] (* Harvey P. Dale, Dec 01 2013 *)
LinearRecurrence[{1,8}, {1,0,4}, 31] (* G. C. Greubel, Feb 04 2023 *)

SageMath

def A100303(n): return (1/2)*int(n==0) + 4*lucas_number1(n-1, 1, -8)
[A100303(n) for n in range(31)] # G. C. Greubel, Feb 04 2023

Formula

a(n) = 0^n/2 + (4/sqrt(33))*( ((1 + sqrt(33))/2)^(n-1) - ((1 - sqrt(33))/2)^(n-1) ).

a(n) = (1/2)*(A015443(n) - A015443(n-1)), n > 0. - Ralf Stephan, Jul 21 2013

E.g.f.: (33 + exp(x/2)*(33*cosh(sqrt(33)*x/2) - sqrt(33)*sinh(sqrt(33)*x/2)))/66. - Stefano Spezia, Sep 08 2022

a(n) = (1/2)*[n=0] + 4*A015443(n). - G. C. Greubel, Feb 04 2023