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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 (list; constant; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..103.

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-August 2, 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-Fran├žois Alcover, Feb 13 2013 *)

PROG

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

CROSSREFS

Cf. A000045, A007598, A190647.

Sequence in context: A305328 A332396 A100124 * A201519 A198507 A021152

Adjacent sequences:  A190645 A190646 A190647 * A190649 A190650 A190651

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

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Last modified July 27 07:50 EDT 2021. Contains 346304 sequences. (Running on oeis4.)