A015536 - OEIS

%I #35 Sep 08 2022 08:44:40

%S 0,1,5,28,155,859,4760,26377,146165,809956,4488275,24871243,137821040,

%T 763718929,4232057765,23451445612,129953401355,720121343611,

%U 3990466922120,22112698641433,122534893973525,679012565791924,3762667510880195,20850375251776747

%N Expansion of x/(1-5*x-3*x^2).

%C This is the Lucas sequence U(5,-3). - _Bruno Berselli_, Jan 09 2013

%H Vincenzo Librandi, <a href="/A015536/b015536.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,3).

%F a(n) = 5*a(n-1) + 3*a(n-2) with n > 1, a(0)=0, a(1)=1.

%F From _Paul Barry_, Jul 20 2004: (Start)

%F a(n) = (5/2 + sqrt(37)/2)^n/sqrt(37) - (5/2 - sqrt(37)/2)^n/sqrt(37).

%F a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)3^k*5^(n-2k-1). (End)

%t Join[{a=0,b=1},Table[c=5*b+3*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)

%t LinearRecurrence[{5, 3}, {0, 1}, 30] (* _Vincenzo Librandi_, Nov 12 2012 *)

%o (Sage) [lucas_number1(n,5,-3) for n in range(0, 22)] # _Zerinvary Lajos_, Apr 24 2009

%o (Magma) [n le 2 select n-1 else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 12 2012

%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1-5*x-3*x^2))) \\ _G. C. Greubel_, Jan 01 2018

%K nonn,easy

%O 0,3

%A _Olivier Gérard_