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A059840
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a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045.
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15
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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
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OFFSET
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1,3
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LINKS
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FORMULA
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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+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
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MAPLE
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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
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Table[If[OddQ[n], Fibonacci[n]Fibonacci[n-1], Fibonacci[n] Fibonacci[n-1]-1], {n, 30}] (* Harvey P. Dale, Apr 20 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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