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%I M3988 #82 Dec 21 2023 11:58:14
%S 0,5,39,272,1869,12815,87840,602069,4126647,28284464,193864605,
%T 1328767775,9107509824,62423800997,427859097159,2932589879120,
%U 20100270056685,137769300517679,944284833567072,6472224534451829,44361286907595735,304056783818718320
%N a(n) = 7*a(n-1) - a(n-2) + 4, with a(0) = 0, a(1) = 5.
%C The values (a(n),x(n)), n >= 2, x(n)=Fibonacci(2*n+2)*Fibonacci(2*n+3)=A081018(n+1), are the integer solutions (a,x) of the equation binomial(x+1,a+1) + binomial(x+2,a+1) = binomial(x+3,a+1). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%C The values (a(n),x(n)), n >= 2 are also the integer solutions (a, x) of the equation x(a+1) = (x-a)(x-a-1) or, equivalently, binomial(x, a) = binomial(x-1, a+1). - _Tomohiro Yamada_, May 30 2018
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A003482/b003482.txt">Table of n, a(n) for n = 0..1000</a>
%H Heiko Harborth, <a href="http://dx.doi.org/10.1007/978-94-015-7801-1_1">Fermat-like binomial equations</a>, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose, California, August 1986, 1-5 (1988).
%H Sébastien Labbé and Jana Lepšová, <a href="https://arxiv.org/abs/2205.02574">A Fibonacci's complement numeration system</a>, arXiv:2205.02574 [cs.FL], 2022.
%H Sébastien Labbé and Jana Lepšová, <a href="https://doi.org/10.1051/ita/2023007">A Fibonacci analogue of the two's complement numeration system</a>, RAIRO-Theor. Inf. Appl. (2023) Vol. 57, No. 12. See p. 16.
%H D. A. Lind, <a href="https://www.fq.math.ca/Scanned/6-3/lind.pdf">The quadratic field Q(sqrt(5)) and a certain diophantine equation</a>, Fibonacci Quart. 6(3) (1968), 86-93.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>
%H David Singmaster, <a href="https://www.fq.math.ca/Scanned/13-4/singmaster.pdf">Repeated binomial coefficients and Fibonacci numbers</a>, Fibonacci Quart. 13 (1973), 295-298.
%H S. M. Tanny and M. Zuker, <a href="http://dx.doi.org/10.1016/0012-365X(74)90073-9">On a unimodal sequence of binomial coefficients</a>, Discrete Math. 9 (1974), 79-89.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F a(n) = Fibonacci(2*n) * Fibonacci(2*n+3).
%F a(n) = Fibonacci(2*n+2)^2 - Fibonacci(2*n+1)^2. - _Gary Detlefs_, Oct 12 2011
%F a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3). - _Vladimir Joseph Stephan Orlovsky_ and _Vincenzo Librandi_, Jan 22 2012
%F a(n) = -4/5 + (sqrt(5)/5 + 2/5)*(7/2 + 3*sqrt(5)/2)^n - (sqrt(5)/5 - 2/5)*(7/2 - 3*sqrt(5)/2)^n. - _Antonio Alberto Olivares_, May 29 2013
%F a(n) = -A206351(-n) for all n in Z. - _Michael Somos_, Jun 26 2018
%F From _Sébastien Labbé_, May 06 2022: (Start)
%F a(n) = Sum_{k=2..2*n+1} Fibonacci(k)^2.
%F a(n) = A001654(2*n+1)-1. (End)
%e G.f. = 5*x + 39*x^2 + 272*x^3 + 1869*x^4 + 12815*x^5 + 87840*x^6 + ... - _Michael Somos_, Jun 26 2018
%p A003482:=z*(-5+z)/(z-1)/(z**2-7*z+1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t LinearRecurrence[{8,-8,1},{0,5,39},30] (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2012 *)
%o (PARI) a(n)=fibonacci(2*n)*fibonacci(2*n+3) \\ _Charles R Greathouse IV_, May 29 2013
%Y Cf. A000045, A001109, A001654, A081018, A206351.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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