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
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).
Programs
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GAP
List([1..30],n->Sum([1..n-2],k->Fibonacci(k)*Fibonacci(k+2))); # Muniru A Asiru, Aug 09 2018
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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
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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
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Mathematica
Table[If[OddQ[n],Fibonacci[n]Fibonacci[n-1],Fibonacci[n] Fibonacci[n-1]-1],{n,30}] (* Harvey P. Dale, Apr 20 2011 *) -
PARI
a(n) = { fibonacci(n)*fibonacci(n-1) - (n%2 == 0) } \\ Harry J. Smith, Jun 29 2009 -
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
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*(-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