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%I #49 Oct 30 2020 08:18:56
%S 2,4,2,6,3,2,0,7,5,1,1,6,7,2,4,1,1,8,7,7,4,1,5,6,9,4,1,2,9,2,6,6,2,0,
%T 3,7,4,3,2,0,2,5,9,7,7,4,5,1,3,8,3,0,9,0,5,1,1,0,1,0,2,8,3,4,5,4,6,6,
%U 1,1,9,3,7,5,1,1,1,9,7,8,6,3,6,8,7,7,5,3,8,9,8,1,5,2,1,5,3,6,3,6,3,7,9,2,1
%N Decimal expansion of sum of reciprocals of squares of Fibonacci numbers.
%C Known to be transcendental. - _Benoit Cloitre_, Jan 07 2006
%C Compare with Sum_{n >= 1} 1/(F(n)^2 + 1) = (5*sqrt(5) - 3)/6 and Sum_{n >= 3} 1/(F(n)^2 - 1) = (43 - 15*sqrt(5))/18. - _Peter Bala_, Nov 19 2019
%C Duverney et al. (1997) proved that this constant is transcendental. - _Amiram Eldar_, Oct 30 2020
%H Richard André-Jeannin, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image">Irrationalité de la somme des inverses de certaines suites récurrentes</a>, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.
%H Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, <a href="http://doi.org/10.3792/pjaa.73.140">Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers</a>, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
%H Michel Waldschmidt, <a href="https://doi.org/10.1007/978-0-387-78510-3_7">Elliptic functions and transcendence</a>, in: Krishnaswami Alladi (ed.), Surveys in number theory, Springer, New York, NY, 2008, pp. 143-188, <a href="http://www.math.jussieu.fr/~miw/articles/pdf/SurveyTrdceEllipt2006.pdf">alternative link</a>. See Corollary 51.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html">Reciprocal Fibonacci Constant</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Sum_{k>=1} 1/F(k)^2 = 2.4263207511672411877... - _Benoit Cloitre_, Jan 07 2006
%e 2.426320751167241187741569...
%t RealDigits[Total[1/Fibonacci[Range[500]]^2],10,120][[1]] (* _Harvey P. Dale_, May 31 2016 *)
%o (PARI) sum(k=1,500,1./fibonacci(k)^2) \\ _Benoit Cloitre_, Jan 07 2006
%Y Cf. A000045, A007598 (squares of Fibonacci numbers).
%Y Cf. A079586, A093540, A105394.
%K cons,easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Apr 04 2005
%E More terms from _Benoit Cloitre_, Jan 07 2006
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