OFFSET
0,3
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 'internal' nodes. (Spectrum : [0^3; 1; (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 (2,7,-8).
FORMULA
a(n) = 2*a(n-1) + 7*a(n-2) - 8*a(n-3).
a(n) = 1/2 + 2^(-n)*(sqrt(33)/132)*((1 + sqrt(33))^(n+1) - (1 - sqrt(33))^(n+1)).
E.g.f.: exp(x)/2 + 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)*(1 + (2*sqrt(2)*i)^n*ChebyshevU(n, -i/(4*sqrt(2)))). - G. C. Greubel, Feb 03 2023
MATHEMATICA
CoefficientList[Series[(1-x-4x^2)/(1-2x-7x^2+8x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 7, -8}, {1, 1, 5}, 40] (* Harvey P. Dale, Oct 05 2012 *)
PROG
(Magma) I:=[1, 1, 5]; [n le 3 select I[n] else 2*Self(n-1) +7*Self(n-2) -8*Self(n-3): n in [1..41]]; // G. C. Greubel, Feb 03 2023
(SageMath)
def A100305(n): return (1/2)*(1 + lucas_number1(n+1, 1, -8))
[A100305(n) for n in range(41)] # G. C. Greubel, Feb 03 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 12 2004
STATUS
approved