A128534 a(n) = Fibonacci(n)*Lucas(n-1).

0, 2, 1, 6, 12, 35, 88, 234, 609, 1598, 4180, 10947, 28656, 75026, 196417, 514230, 1346268, 3524579, 9227464, 24157818, 63245985, 165580142, 433494436, 1134903171, 2971215072, 7778742050, 20365011073, 53316291174, 139583862444, 365435296163, 956722026040, 2504730781962

Offset: 0

Author

Axel Harvey, Mar 08 2007

Keywords

• easy
• nonn

Comments

Generally, F(n)*L(n+k) = F(2*n + k) + F(k)*(-1)^(n+1). If k=0 the sequence is A001906; if k=1 it is A081714.

a(n) is the maximum area of a quadrilateral with lengths of sides in order F(n), F(n), L(n-1), L(n-1) for n>1. - J. M. Bergot, Jan 28 2016

Can be obtained (up to signs) by setting x = F(n)/F(n+1) in g.f. for Lucas numbers - see Pongsriiam. - N. J. A. Sloane, Mar 23 2017

Examples

			a(5) = 35 because F(5)*L(4) = 5*7.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2370
• Prapanpong Pongsriiam, Integral Values of the Generating Functions of Fibonacci and Lucas Numbers, College Math. J., 48 (No. 2 2017), pp 97ff.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A000032, A000045, A001906, A081714, A128533, A128535.

Programs

Magma

[Fibonacci(n)*Lucas(n-1): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Maple

seq(combinat:-fibonacci(2*n-1)+(-1)^(n+1),n=0..50); # Robert Israel, Jan 28 2016

Mathematica

Table[Fibonacci[n] LucasL[n - 1], {n, 0, 31}] (* Michael De Vlieger, Jan 29 2016 *)

PARI

concat( 0, Vec(-x*(-2+3*x)/((1+x)*(x^2-3*x+1)) + O(x^40))) \\ Michel Marcus, Jan 28 2016

Formula

a(n) = F(2*n - 1) + (-1)^(n+1), assuming F(0)=0 and L(0)=2.

From R. J. Mathar, Apr 16 2009: (Start)

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).

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

a(n) = (2^(-1-n)*(-5*(-1)^n*2^(1+n) - (-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5))))/5. - Colin Barker, Apr 05 2016

a(n+1) = A081714(n) + 2*(-1)^n. - A.H.M. Smeets, Feb 26 2022

Extensions

More terms from Michel Marcus, Jan 28 2016