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A080097 a(n) = Fibonacci(n+2)^2 - 1. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

S. 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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

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

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 *)

PROG

(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

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

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

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

CROSSREFS

Equals A007598(n+2) - 1.

Cf. A000032, A000045, A064831, A059840.

Sequence in context: A323278 A026556 A096001 * A096886 A176904 A056332

Adjacent sequences:  A080094 A080095 A080096 * A080098 A080099 A080100

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

Edited by Ralf Stephan, May 15 2005

STATUS

approved

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Last modified April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)