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%I #12 Jan 05 2025 19:51:42
%S 5,8,0,0,0,4,7,3,9,5,0,7,7,7,0,6,8,0,0,6,7,4,7,0,9,8,1,8,9,5,5,2,2,8,
%T 0,2,6,9,8,5,0,1,2,6,0,9,6,4,6,1,6,3,9,0,1,5,7,7,5,6,1,0,0,1,7,7,6,7,
%U 3,7,5,7,5,2,1,9,9,7,8,4,8,9,4,9,2,1,0,4,4,7,8,6,6,9,4,0,2,2,3,7,1,4,1,1,5
%N Decimal expansion of Sum_{k>=1} (-1)^(k+1)*k/Fibonacci(2*k).
%H Daniel Duverney and Iekata Shiokawa, <a href="https://doi.org/10.1063/1.2841912">On series involving Fibonacci and Lucas numbers I</a>, AIP Conference Proceedings, Vol. 976, No. 1. American Institute of Physics, 2008, pp. 62-76.
%H Derek Jennings, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-1/jennings.pdf">On reciprocals of Fibonacci and Lucas numbers</a>, Fibonacci Quarterly, Vol. 32, No. 1 (1994), pp. 18-21.
%F Equals Sum_{k>=1} (-1)^(k+1)*k/A001906(k).
%F Equals (1/sqrt(5)) * Sum_{k>=1} 1/Fibonacci(2*k-1)^2 (Jennings, 1994).
%e 0.58000473950777068006747098189552280269850126096461...
%t RealDigits[Sum[(-1)^(k+1)*k/Fibonacci[2*k], {k, 1, 300}], 10, 100][[1]]
%o (PARI) sumalt(k=1, (-1)^(k+1)*k/fibonacci(2*k)) \\ _Michel Marcus_, Sep 10 2022
%Y Cf. A000045, A001519, A001906, A081068, A357053.
%Y Cf. A079586, A153386, A153387, A158933, A265288.
%K nonn,cons,changed
%O 0,1
%A _Amiram Eldar_, Sep 10 2022