OFFSET

0,2

COMMENTS

Borwein et al. express the sum in terms of theta function values.

Compare with Sum_{k >= 1} (-1)^(k+1)/(F(k)^2 + 1) = (3 - sqrt(5))/6 and Sum_{k >= 3} (-1)^(k+1)/(F(k)^2 - 1) = (11 - 3*sqrt(5))/18. - Peter Bala, Nov 13 2019

Duverney (1997) proved that this constant does not belong to the quadratic number field Q(sqrt(5)), and Duverney et al. (1998) proved that it is transcendental. - Amiram Eldar, Oct 30 2020

REFERENCES

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 203.

LINKS

Daniel Duverney, Some arithmetical consequences of Jacobi's triple product identity, Math. Proc. Camb. Phil. Soc., Vol. 122, No. 3 (1997), pp. 393-399.

Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Jacobi's theta series and related results, in: Kalman Gyoery, Attila Pethoe and Vera T. Sos (eds.), Number Theory-Diophantine, Computational and Algebraic Aspects, Proceedings of the International Conference held in Eger, Hungary, July 29-Aug 02 1996, De Gruyter, 1998, pp. 157-168.

EXAMPLE

0.16753929845562511832413984100914483853736687...

MAPLE

with(combinat): evalf[105](add((-1)^(k+1)/fibonacci(k)^2, k=1..500)); # Nathaniel Johnston, May 24 2011

MATHEMATICA

Clear[f]; f[n_] := f[n] = RealDigits[ Sum[(-1)^(k+1)/Fibonacci[k]^2, {k, 1, n}], 10, 104] // First; f[n=100]; While[f[n] != f[n-100], n = n+100]; f[n] (* Jean-François Alcover, Feb 13 2013 *)

PROG

(PARI) suminf(k=1, (-1)^(k+1)/fibonacci(k)^2) \\ Michel Marcus, Nov 19 2019

CROSSREFS

KEYWORD

cons,nonn

AUTHOR

N. J. A. Sloane, May 16 2011

EXTENSIONS

a(49) corrected and more terms from Nathaniel Johnston, May 24 2011

Typo in Name and Maple program corrected by Peter Bala, Nov 13 2019

STATUS

approved