A105393 - OEIS

A105393

Decimal expansion of sum of reciprocals of squares of Fibonacci numbers.

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, 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, 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

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OFFSET

1,1

COMMENTS

Known to be transcendental. - Benoit Cloitre, Jan 07 2006

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

Duverney et al. (1997) proved that this constant is transcendental. - Amiram Eldar, Oct 30 2020

LINKS

Table of n, a(n) for n=1..105.

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.

Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.

Michel Waldschmidt, Elliptic functions and transcendence, in: Krishnaswami Alladi (ed.), Surveys in number theory, Springer, New York, NY, 2008, pp. 143-188, alternative link. See Corollary 51.

Eric Weisstein's World of Mathematics, Fibonacci Number.

Eric Weisstein's World of Mathematics, Lucas Number.

Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.

Index entries for transcendental numbers

FORMULA

Equals Sum_{k>=1} 1/F(k)^2 = 2.4263207511672411877... - Benoit Cloitre, Jan 07 2006

EXAMPLE

2.426320751167241187741569...

MATHEMATICA

RealDigits[Total[1/Fibonacci[Range[500]]^2], 10, 120][[1]] (* Harvey P. Dale, May 31 2016 *)

PROG

(PARI) sum(k=1, 500, 1./fibonacci(k)^2) \\ Benoit Cloitre, Jan 07 2006