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 extremity nodes. a(n) counts closed walks of length n at each of the extremity 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) = 3/4 + (1/(4*sqrt(33)))*(((1 + sqrt(33))/2)^(n+1) - ((1 - sqrt(33))/2)^(n+1)).
E.g.f.: 3*exp(x)/4 + exp(x/2)*(33*cosh(sqrt(33)*x/2) + sqrt(33)*sinh(sqrt(33)*x/2))/132. - Stefano Spezia, Sep 08 2022
a(n) = (1/4)*(3 + A015443(n+1)). - G. C. Greubel, Feb 04 2023
MATHEMATICA
LinearRecurrence[{2, 7, -8}, {1, 1, 3}, 29] (* Stefano Spezia, Sep 08 2022 *)
PROG
(Magma) I:=[1, 1, 3]; [n le 3 select I[n] else 2*Self(n-1) +7*Self(n-2) -8*Self(n-3): n in [1..31]]; // G. C. Greubel, Feb 04 2023
(SageMath)
def A100302(n): return (3 + lucas_number1(n+1, 1, -8))/4
[A100302(n) for n in range(31)] # G. C. Greubel, Feb 04 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 12 2004
STATUS
approved