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A059840 a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
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.
FORMULA
G.f.: (x^3)*(2-x)/((1-x^2)*(1-3*x+x^2)), with a(0):=0. 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) { b=0; f=1; for (n=1, 500, a=f*b; if (frac(n/2)==0, a--); write("b059840.txt", n, " ", a); a=f + b; b=f; f=a; ) } \\ 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
Sequence in context: A148341 A258121 A242823 * A280064 A148342 A148343
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 26 2001
STATUS
approved

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Last modified March 19 02:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)