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%I #50 Nov 15 2023 00:55:25
%S 0,3,8,24,63,168,440,1155,3024,7920,20735,54288,142128,372099,974168,
%T 2550408,6677055,17480760,45765224,119814915,313679520,821223648,
%U 2149991423,5628750624,14736260448,38580030723,101003831720
%N a(n) = Fibonacci(n+2)^2 - 1.
%C a(n), a(n)+1 and a(n)+2 are consecutive members of A049997.
%H G. C. Greubel, <a href="/A080097/b080097.txt">Table of n, a(n) for n = 0..1000</a>
%H Sergio 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F If n is odd, then a(n) = F(n+1)*F(n+3) = F(n)*F(n+4) - 2, else a(n) = F(n)*F(n+4) = F(n+1)*F(n+3) - 2, where F(n) = Fibonacci numbers (A000045).
%F a(n) = (Lucas(2*n+4) - 2*(-1)^n - 5)/5.
%F O.g.f.: x*(3-x)/((1-x^2)*(1-3*x+x^2)) (see a comment on A080144). - _Wolfdieter Lang_, Jul 30 2012
%F a(n) = Sum_{k=1..n} F(k+3)*F(k) = A027941(n) + 2*A001654(n), n>=0. - _Wolfdieter Lang_, Jul 27 2012
%F Sum_{n>=1} 1/a(n) = (43 - 15*sqrt(5))/18 = 29/9 - 5*phi/3, where phi is the golden ratio (A001622). - _Amiram Eldar_, Oct 20 2020
%F a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - _Joerg Arndt_, Nov 13 2023
%t CoefficientList[Series[(3x+2x^2-x^3)/(1-x^2)(1-2x-2x^2+x^3)), {x, 0, 30}], x]
%t Table[Fibonacci[n+2]^2-1,{n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 03 2011 *)
%o (Maxima) A080097(n):=fib(n+2)^2-1$ makelist(A080097(n),n,0,30); /* _Martin Ettl_, Nov 13 2012 */
%o (PARI) a(n)=fibonacci(n+2)^2-1 \\ _Charles R Greathouse IV_, Feb 06 2013
%o (Magma) [Fibonacci(n+2)^2 -1: n in [0..30]]; // _G. C. Greubel_, Jul 23 2019
%o (Sage) [fibonacci(n+2)^2 -1 for n in (0..30)] # _G. C. Greubel_, Jul 23 2019
%o (GAP) List([0..30], n-> Fibonacci(n+2)^2 -1); # _G. C. Greubel_, Jul 23 2019
%Y Equals A007598(n+2) - 1.
%Y Cf. A000032, A000045, A001622, A049997, A059840, A064831, A080144.
%K easy,nonn
%O 0,2
%A Mario Catalani (mario.catalani(AT)unito.it), Jan 29 2003
%E Edited by _Ralf Stephan_, May 15 2005
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