%I #54 Jun 25 2020 19:32:08

%S 2,5,29,155,833,4475,24041,129155,693857,3727595,20025689,107583635,

%T 577969553,3105015035,16681014281,89615101475,481437535937,

%U 2586417882635,13894964485049,74647658190515,401028219922673,2154436415994395,11574238519817321,62180065431075395

%N a(n) is the trace of the n-th power of the 2 X 2 matrix [1 2; 3 4].

%H Colin Barker, <a href="/A327765/b327765.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,2).

%F a(n) = trace(M^n) where M is [1, 2; 3, 4].

%F From _Colin Barker_, Sep 27 2019: (Start)

%F G.f.: (2 - 5*x) / (1 - 5*x - 2*x^2).

%F a(n) = 5*a(n-1) + 2*a(n-2) for n > 1.

%F a(n) = ((5-sqrt(33))/2)^n + ((5+sqrt(33))/2)^n.

%F (End)

%p a:= n-> (<<0|1>, <2|5>>^n. <<2, 5>>)[1, 1]:

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Oct 07 2019

%t CoefficientList[Series[(2 - 5 x)/(1 - 5 x - 2 x^2), {x, 0, 22}], x] (* _Michael De Vlieger_, Sep 27 2019 *)

%t LinearRecurrence[{5,2},{2,5},30] (* _Harvey P. Dale_, Jun 25 2020 *)

%o (PARI) a(n)={trace([1,2;3,4]^n)} \\ _Andrew Howroyd_, Sep 24 2019

%o (PARI) Vec((2 - 5*x) / (1 - 5*x - 2*x^2) + O(x^25)) \\ _Colin Barker_, Sep 27 2019

%Y Cf. A100638, A124610, A233020.

%K nonn,easy

%O 0,1

%A _Adolf Cusmariu_, Sep 24 2019