A097512 - OEIS

%I #43 Jan 01 2024 11:27:45

%S 11,15,34,87,227,594,1555,4071,10658,27903,73051,191250,500699,

%T 1310847,3431842,8984679,23522195,61581906,161223523,422088663,

%U 1105042466,2893038735,7574073739,19829182482,51913473707,135911238639

%N a(n) = 6*Lucas(2n) - Fibonacci(2n+2).

%C Sequence relates bisections of Lucas and Fibonacci numbers.

%H Vincenzo Librandi, <a href="/A097512/b097512.txt">Table of n, a(n) for n = 0..200</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).

%F a(n) = 8*Lucas(2n) - Lucas(2n+2) - 2*Fibonacci(2n-1) = 8*A005248(n) - A005248(n+1) - 2*A001519(n).

%F a(n+1)/a(n) approaches the golden ratio phi + 1 = (3+sqrt(5))/2.

%F From _Philippe Deléham_, Nov 16 2008: (Start)

%F a(n) = 3*a(n-1) - a(n-2) with a(0)=11 and a(1)=15.

%F G.f.: (11-18*x)/(1-3*x+x^2). (End)

%F a(n) = 4*Fibonacci(2n+1) + 7*Fibonacci(2n-1) = 4*Lucas(2n) + 3*Fibonacci(2n-1). - _Ron Knott_, Jul 01 2013

%F E.g.f.: exp(3*x/2)*( 11*cosh(sqrt(5)*x/2) - (3/sqrt(5))*sinh(sqrt(5)*x/2) ). - _G. C. Greubel_, Jan 19 2020

%p f:=fibonacci; f[combinat]; seq(4*f(2*n+1)+7*f(2*n-1), n=0..30); # _G. C. Greubel_, Jan 19 2020

%t Table[6LucasL[2n]-Fibonacci[2n+2],{n,0,30}] (* or *) LinearRecurrence[ {3,-1},{11,15},30] (* _Harvey P. Dale_, Oct 02 2011 *)

%o (Magma) [8*Lucas(2*n) - Lucas(2*n+2) - 2*Fibonacci(2*n-1): n in [0..30]]; // _Vincenzo Librandi_, Oct 03 2011

%o (PARI) vector(31, n, 4*fibonacci(2*n-1) +7*fibonacci(2*n-3) ) \\ _G. C. Greubel_, Jan 19 2020

%o (Sage) [6*lucas_number2(2*n,1,-1) - fibonacci(2*n+2) for n in (0..30)] # _G. C. Greubel_, Jan 19 2020

%o (GAP) List([0..30], n-> 6*Lucas(1,-1,2*n)[2] - Fibonacci(2*n+2) ); # _G. C. Greubel_, Jan 19 2020

%Y Cf. A000032, A000045, A005248, A022133.

%K nonn,easy

%O 0,1

%A _Creighton Dement_, Aug 26 2004

%E New definition from _Ralf Stephan_, Dec 01 2004