A059840 a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045.

0, 0, 2, 5, 15, 39, 104, 272, 714, 1869, 4895, 12815, 33552, 87840, 229970, 602069, 1576239, 4126647, 10803704, 28284464, 74049690, 193864605, 507544127, 1328767775, 3478759200, 9107509824, 23843770274, 62423800997, 163427632719, 427859097159, 1120149658760, 2932589879120

Offset

1,3

Links

Harry J. Smith, Table of n, a(n) for n = 1..500

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.

H. Ohtsuka and S. Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 153-159.

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

Formula

G.f.: x^3*(2 - 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-2} F(k)*F(k+2). - Alexander Adamchuk, May 17 2007

a(n+2) = (3*A001654(n) + A027941(n))/2, n >= 0. - Wolfdieter Lang, Jul 21 2012

a(n+2) = (3*(-1)^(n+1) - 5 + 2*Lucas(2*n + 3))/10, n >= 0. - Ehren Metcalfe, Aug 21 2017

a(n) = floor(1/(Sum_{k>=n} 1/Fibonacci(k)^2)) [Ohtsuka and Nakamura]. - Michel Marcus, Aug 09 2018

For n > 2, 2 * A000217(a(n)) = A228873(n-2). - Diego Rattaggi, Jan 27 2021

Maple

seq(coeff(series(x^3*(2-x)/((1-x^2)*(1-3*x+x^2)), x, n+1), x, n), n=1..30); # Muniru A Asiru, Aug 09 2018

Mathematica

Table[If[OddQ[n], Fibonacci[n]Fibonacci[n-1], Fibonacci[n] Fibonacci[n-1]-1], {n, 30}]  (* Harvey P. Dale, Apr 20 2011 *)

Prog

(PARI) a(n) = { fibonacci(n)*fibonacci(n-1) - (n%2 == 0) } \\ Harry J. Smith, Jun 29 2009
(GAP) List([1..30], n->Sum([1..n-2], k->Fibonacci(k)*Fibonacci(k+2))); # Muniru A Asiru, Aug 09 2018
(Magma) F:=Fibonacci; [(n mod 2) eq 0 select F(n)*F(n-1)-1 else F(n)*F(n-1): n in [1..30]]; // G. C. Greubel, Jul 23 2019
(Sage) a=(x^3*(2-x)/((1-x^2)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 23 2019

Crossrefs

Cf. A000045, A001654, A059248, A064831, A119996.

Sequence in context: A148341 A258121 A242823 * A280064 A148342 A148343

Adjacent sequences: A059837 A059838 A059839 * A059841 A059842 A059843

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 26 2001

Status

approved