OFFSET
0,1
COMMENTS
Sequence relates bisections of Lucas and Fibonacci numbers.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).
FORMULA
a(n+1)/a(n) approaches the golden ratio phi + 1 = (3+sqrt(5))/2.
From Philippe Deléham, Nov 16 2008: (Start)
a(n) = 3*a(n-1) - a(n-2) with a(0)=11 and a(1)=15.
G.f.: (11-18*x)/(1-3*x+x^2). (End)
a(n) = 4*Fibonacci(2n+1) + 7*Fibonacci(2n-1) = 4*Lucas(2n) + 3*Fibonacci(2n-1). - Ron Knott, Jul 01 2013
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
MAPLE
f:=fibonacci; f[combinat]; seq(4*f(2*n+1)+7*f(2*n-1), n=0..30); # G. C. Greubel, Jan 19 2020
MATHEMATICA
Table[6LucasL[2n]-Fibonacci[2n+2], {n, 0, 30}] (* or *) LinearRecurrence[ {3, -1}, {11, 15}, 30] (* Harvey P. Dale, Oct 02 2011 *)
PROG
(Magma) [8*Lucas(2*n) - Lucas(2*n+2) - 2*Fibonacci(2*n-1): n in [0..30]]; // Vincenzo Librandi, Oct 03 2011
(PARI) vector(31, n, 4*fibonacci(2*n-1) +7*fibonacci(2*n-3) ) \\ G. C. Greubel, Jan 19 2020
(Sage) [6*lucas_number2(2*n, 1, -1) - fibonacci(2*n+2) for n in (0..30)] # G. C. Greubel, Jan 19 2020
(GAP) List([0..30], n-> 6*Lucas(1, -1, 2*n)[2] - Fibonacci(2*n+2) ); # G. C. Greubel, Jan 19 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Aug 26 2004
EXTENSIONS
New definition from Ralf Stephan, Dec 01 2004
STATUS
approved