The OEIS is supported by the many generous donors to the OEIS Foundation.

A190648
Decimal expansion of Sum_{k>=1} (-1)^(k+1)/Fibonacci(k)^2.
1
1, 6, 7, 5, 3, 9, 2, 9, 8, 4, 5, 5, 6, 2, 5, 1, 1, 8, 3, 2, 4, 1, 3, 9, 8, 4, 1, 0, 0, 9, 1, 4, 4, 8, 3, 8, 5, 3, 7, 3, 6, 6, 8, 7, 1, 5, 9, 9, 2, 8, 3, 7, 9, 8, 4, 3, 3, 9, 0, 0, 0, 6, 9, 6, 0, 8, 6, 8, 0, 2, 7, 3, 3, 2, 2, 2, 3, 3, 7, 0, 4, 5, 0, 8, 9, 7, 7, 0, 8, 7, 2, 6, 5, 2, 9, 7, 4, 7, 2, 8, 2, 3, 2, 8
OFFSET
0,2
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.
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 20 09:57 EDT 2024. Contains 376068 sequences. (Running on oeis4.)