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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.

%I A059840 #54 Jan 05 2025 19:51:36
%S A059840 0,0,2,5,15,39,104,272,714,1869,4895,12815,33552,87840,229970,602069,
%T A059840 1576239,4126647,10803704,28284464,74049690,193864605,507544127,
%U A059840 1328767775,3478759200,9107509824,23843770274,62423800997,163427632719,427859097159,1120149658760,2932589879120
%N A059840 a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045.
%H A059840 Harry J. Smith, <a href="/A059840/b059840.txt">Table of n, a(n) for n = 1..500</a>
%H A059840 S. Falcon, <a href="https://www.researchgate.net/publication/298789400_On_the_Sequences_of_Products_of_Two_k-Fibonacci_Numbers">On the Sequences of Products of Two k-Fibonacci Numbers</a>, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
%H A059840 H. Ohtsuka and S. Nakamura, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/46_47-2/Ohtsuka_Nakamura.pdf">On the sum of reciprocal sums of Fibonacci numbers</a>, Fibonacci Quart. 46/47 (2008/2009), 153-159.
%H A059840 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F A059840 G.f.: x^3*(2 - x)/((1 - x^2)*(1 - 3*x + x^2)). See a comment on A080144. - _Wolfdieter Lang_, Jul 30 2012
%F A059840 a(n) = Sum_{k=1..n-2} F(k)*F(k+2). - _Alexander Adamchuk_, May 17 2007
%F A059840 a(n+2) = (3*A001654(n) + A027941(n))/2, n >= 0. - _Wolfdieter Lang_, Jul 21 2012
%F A059840 a(n+2) = (3*(-1)^(n+1) - 5 + 2*Lucas(2*n + 3))/10, n >= 0. - _Ehren Metcalfe_, Aug 21 2017
%F A059840 a(n) = floor(1/(Sum_{k>=n} 1/Fibonacci(k)^2)) [Ohtsuka and Nakamura]. - _Michel Marcus_, Aug 09 2018
%F A059840 For n > 2, 2 * A000217(a(n)) = A228873(n-2). - _Diego Rattaggi_, Jan 27 2021
%p A059840 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
%t A059840 Table[If[OddQ[n],Fibonacci[n]Fibonacci[n-1],Fibonacci[n] Fibonacci[n-1]-1],{n,30}]  (* _Harvey P. Dale_, Apr 20 2011 *)
%o A059840 (PARI) a(n) = { fibonacci(n)*fibonacci(n-1) - (n%2 == 0) } \\ _Harry J. Smith_, Jun 29 2009
%o A059840 (GAP) List([1..30],n->Sum([1..n-2],k->Fibonacci(k)*Fibonacci(k+2))); # _Muniru A Asiru_, Aug 09 2018
%o A059840 (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
%o A059840 (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
%Y A059840 Cf. A000045, A001654, A059248, A064831, A119996.
%K A059840 nonn,easy
%O A059840 1,3
%A A059840 _N. J. A. Sloane_, Feb 26 2001