A080097 a(n) = Fibonacci(n+2)^2 - 1.

0, 3, 8, 24, 63, 168, 440, 1155, 3024, 7920, 20735, 54288, 142128, 372099, 974168, 2550408, 6677055, 17480760, 45765224, 119814915, 313679520, 821223648, 2149991423, 5628750624, 14736260448, 38580030723, 101003831720

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Jan 29 2003

Keywords

• easy
• nonn

Comments

a(n), a(n)+1 and a(n)+2 are consecutive members of A049997.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Sergio Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Crossrefs

Equals A007598(n+2) - 1.

Cf. A000032, A000045, A001622, A049997, A059840, A064831, A080144.

Programs

GAP

List([0..30], n-> Fibonacci(n+2)^2 -1); # G. C. Greubel, Jul 23 2019

Magma

[Fibonacci(n+2)^2 -1: n in [0..30]]; // G. C. Greubel, Jul 23 2019

Mathematica

CoefficientList[Series[(3x+2x^2-x^3)/((1-x^2)(1-2x-2x^2+x^3)), {x, 0, 30}], x]
Table[Fibonacci[n+2]^2-1,{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)
LinearRecurrence[{3,0,-3,1},{0,3,8,24},40] (* Harvey P. Dale, Nov 23 2024 *)

Maxima

A080097(n):=fib(n+2)^2-1$ makelist(A080097(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

PARI

a(n)=fibonacci(n+2)^2-1 \\ Charles R Greathouse IV, Feb 06 2013

Sage

[fibonacci(n+2)^2 -1 for n in (0..30)] # G. C. Greubel, Jul 23 2019

Formula

If n is odd, then a(n) = F(n+1)*F(n+3) = F(n)*F(n+4) - 2, else a(n) = F(n)*F(n+4) = F(n+1)*F(n+3) - 2, where F(n) = Fibonacci numbers (A000045).

a(n) = (Lucas(2*n+4) - 2*(-1)^n - 5)/5.

O.g.f.: x*(3-x)/((1-x^2)*(1-3*x+x^2)) (see a comment on A080144). - Wolfdieter Lang, Jul 30 2012

a(n) = Sum_{k=1..n} F(k+3)*F(k) = A027941(n) + 2*A001654(n), n>=0. - Wolfdieter Lang, Jul 27 2012

Sum_{n>=1} 1/a(n) = (43 - 15*sqrt(5))/18 = 29/9 - 5*phi/3, where phi is the golden ratio (A001622). - Amiram Eldar, Oct 20 2020

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - Joerg Arndt, Nov 13 2023

Extensions

Edited by Ralf Stephan, May 15 2005