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A260304 a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.

%I A260304 #36 Sep 08 2022 08:46:13
%S A260304 2,3,5,10,25,75,250,875,3125,11250,40625,146875,531250,1921875,
%T A260304 6953125,25156250,91015625,329296875,1191406250,4310546875,
%U A260304 15595703125,56425781250,204150390625,738623046875,2672363281250,9668701171875,34981689453125,126564941406250
%N A260304 a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.
%C A260304 Lim_{n -> infinity} a(n + 1)/a(n) = 2 + phi = 3.6180339887..., where phi is the golden ratio (A001622).
%H A260304 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).
%F A260304 G.f.: (2 - 7*x)/(1 - 5*x + 5*x^2).
%F A260304 a(n) = ((5 + 2*sqrt(5))*((5 - sqrt(5))/2)^n + (5 - 2*sqrt(5))*((5 + sqrt(5))/2)^n)/5.
%F A260304 a(n) = 2*A030191(n) - 7*A030191(n-1). - _Bruno Berselli_, Nov 23 2015
%t A260304 Table[((5 + 2 Sqrt[5]) ((5 - Sqrt[5])/2)^n + (5 - 2 Sqrt[5]) ((5 + Sqrt[5])/2)^n)/5, {n, 0, 30}]
%t A260304 RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 5 a[n - 1] - 5 a[n - 2]}, a, {n, 0, 30}] (* _Bruno Berselli_, Nov 23 2015 *)
%o A260304 (Magma) [n le 2 select n+1 else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 23 2015
%o A260304 (PARI) a(n)=([0,1; -5,5]^n*[2;3])[1,1] \\ _Charles R Greathouse IV_, Jul 26 2016
%Y A260304 Cf. A020876, A030191, A098111.
%Y A260304 Cf. A093129: initial values 1,2; A081567: initial values 1,3.
%K A260304 nonn,easy
%O A260304 0,1
%A A260304 _Ilya Gutkovskiy_, Nov 21 2015
%E A260304 Edited by _Bruno Berselli_, Nov 23 2015