A327765 a(n) is the trace of the n-th power of the 2 X 2 matrix [1 2; 3 4].
2, 5, 29, 155, 833, 4475, 24041, 129155, 693857, 3727595, 20025689, 107583635, 577969553, 3105015035, 16681014281, 89615101475, 481437535937, 2586417882635, 13894964485049, 74647658190515, 401028219922673, 2154436415994395, 11574238519817321, 62180065431075395
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (5,2).
Programs
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Maple
a:= n-> (<<0|1>, <2|5>>^n. <<2, 5>>)[1, 1]: seq(a(n), n=0..23); # Alois P. Heinz, Oct 07 2019
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Mathematica
CoefficientList[Series[(2 - 5 x)/(1 - 5 x - 2 x^2), {x, 0, 22}], x] (* Michael De Vlieger, Sep 27 2019 *) LinearRecurrence[{5,2},{2,5},30] (* Harvey P. Dale, Jun 25 2020 *) -
PARI
a(n)={trace([1,2;3,4]^n)} \\ Andrew Howroyd, Sep 24 2019 -
PARI
Vec((2 - 5*x) / (1 - 5*x - 2*x^2) + O(x^25)) \\ Colin Barker, Sep 27 2019
Formula
a(n) = trace(M^n) where M is [1, 2; 3, 4].
From Colin Barker, Sep 27 2019: (Start)
G.f.: (2 - 5*x) / (1 - 5*x - 2*x^2).
a(n) = 5*a(n-1) + 2*a(n-2) for n > 1.
a(n) = ((5-sqrt(33))/2)^n + ((5+sqrt(33))/2)^n.
(End)