A164581 - OEIS

%I #30 Feb 14 2024 10:06:59

%S 1,2,11,57,296,1537,7981,41442,215191,1117397,5802176,30128277,

%T 156443561,812346082,4218173971,21903215937,113734253656,590574484217,

%U 3066606674741,15923607857922,82684645964351,429346837679677,2229418834362736,11576441009493357

%N a(n) = 5*a(n - 1) + a(n - 2), with a(0)=1, a(1)=2.

%H Vincenzo Librandi, <a href="/A164581/b164581.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 (5,1).

%F a(n) = 5*a(n-1)+a(n-2) = A052918(n)-3*A052918(n-1).

%F G.f.: (1-3*x)/(1-5*x-x^2).

%F a(n) = A052918(n) + A015449(n). - _R. J. Mathar_, Jul 06 2012

%F a(n) = (2^(-1-n)*((5-sqrt(29))^n*(1+sqrt(29))+(-1+sqrt(29))*(5+sqrt(29))^n))/sqrt(29). - _Colin Barker_, Oct 13 2015

%F a(n) = Sum_{k=0..n-2} A168561(n-2,k)*5^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*5^k, n>0. - _R. J. Mathar_, Feb 14 2024

%F a(n) = A052918(n) -3*A052918(n-1). - _R. J. Mathar_, Feb 14 2024

%t LinearRecurrence[{5, 1}, {1, 2}, 40] (* or *) Rest[CoefficientList[Series [x (1 - 3 x) / (1 - 5 x - x^2), {x, 0, 40}], x]] (* _Harvey P. Dale_, May 02 2011 *)

%o (Magma) [ n le 2 select (n) else 5*Self(n-1)+Self(n-2): n in [1..25] ]; // _Vincenzo Librandi_, Sep 12 2013

%o (PARI) Vec((1-3*x)/(1-5*x-x^2) + O(x^40)) \\ _Colin Barker_, Oct 13 2015

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Aug 17 2009