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A190648
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Decimal expansion of Sum_{k>=1} (-1)^(k+1)/Fibonacci(k)^2.
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1
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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.
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EXAMPLE
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0.16753929845562511832413984100914483853736687...
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MAPLE
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with(combinat): evalf[105](add((-1)^(k+1)/fibonacci(k)^2, k=1..500)); # Nathaniel Johnston, May 24 2011
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MATHEMATICA
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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 *)
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PROG
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(PARI) suminf(k=1, (-1)^(k+1)/fibonacci(k)^2) \\ Michel Marcus, Nov 19 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Typo in Name and Maple program corrected by Peter Bala, Nov 13 2019
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STATUS
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approved
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