login
a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045.
15

%I #47 Sep 08 2022 08:45:03

%S 0,0,2,5,15,39,104,272,714,1869,4895,12815,33552,87840,229970,602069,

%T 1576239,4126647,10803704,28284464,74049690,193864605,507544127,

%U 1328767775,3478759200,9107509824,23843770274,62423800997,163427632719

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

%H Harry J. Smith, <a href="/A059840/b059840.txt">Table of n, a(n) for n = 1..500</a>

%H 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 H. Ohtsuka and S. Nakamura, <a href="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.

%F 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

%F a(n) = Sum_{k=1..n-2} F(k)*F(k+2). - _Alexander Adamchuk_, May 17 2007

%F a(n+2) = (3*A001654(n) + A027941(n))/2, n >= 0. - _Wolfdieter Lang_, Jul 21 2012

%F a(n+2) = (3*(-1)^(n+1) - 5 + 2*Lucas(2*n + 3))/10, n >= 0. - _Ehren Metcalfe_, Aug 21 2017

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

%F For n > 2, 2 * A000217(a(n)) = A228873(n-2). - _Diego Rattaggi_, Jan 27 2021

%p 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 Table[If[OddQ[n],Fibonacci[n]Fibonacci[n-1],Fibonacci[n] Fibonacci[n-1]-1],{n,30}] (* _Harvey P. Dale_, Apr 20 2011 *)

%o (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

%o (GAP) List([1..30],n->Sum([1..n-2],k->Fibonacci(k)*Fibonacci(k+2))); # _Muniru A Asiru_, Aug 09 2018

%o (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 (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 Cf. A000045, A001654, A059248, A064831, A119996.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Feb 26 2001