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%I #49 Feb 09 2023 10:24:51
%S 2,8,9,1,4,4,6,4,8,5,7,0,6,7,1,5,8,3,1,1,2,3,0,5,5,0,9,6,1,5,7,2,9,1,
%T 6,6,9,5,4,8,8,1,9,5,1,5,8,9,6,9,1,3,6,0,0,2,5,0,2,6,4,8,5,0,6,3,0,3,
%U 5,7,6,1,7,3,8,8,6,4,5,5,1,5,8,2,4,1,1,5,8,3,1,8,2,8,5
%N Decimal expansion of Sum_{n>=1} ((-1)^(n+1))/F(n) where F(n) is the n-th Fibonacci number A000045(n).
%C André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)). - _Amiram Eldar_, Oct 30 2020
%H Richard André-Jeannin, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image">Irrationalité de la somme des inverses de certaines suites récurrentes</a>, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
%H Yohei Tachiya, <a href="https://projecteuclid.org/euclid.tjm/1244208475">Irrationality of certain Lambert series</a>, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Reciprocal Fibonacci Constant</a>.
%F Equals sqrt(5) * Sum_{k>=0} (-1)^k/(phi^(2*k+1) + (-1)^k), where phi is the golden ratio (A001622). - _Amiram Eldar_, Oct 04 2020
%F Equals A153387 - A153386. - _Joerg Arndt_, Oct 04 2020
%F Equals 1 - A324007. - _Amiram Eldar_, Feb 09 2023
%e 0.2891446485706715831123055096157291669...
%p with(combinat, fibonacci):Digits:=100:s:=0:for n from 1 to 2000 do: a1:=fibonacci(n):s:=s+evalf(1/a1)*(-1)^(n+1):od:print(s):
%t digits = 95; NSum[(-1)^(n+1)*(1/Fibonacci[n]), {n, 1, Infinity}, WorkingPrecision -> digits+1, NSumTerms -> digits] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Jan 28 2014 *)
%o (PARI) -sumalt(n=1,(-1)^n/fibonacci(n)) \\ _Charles R Greathouse IV_, Oct 03 2016
%Y Cf. A000045, A001622, A079586.
%Y Cf. A153386, A153387, A324007.
%K nonn,cons
%O 0,1
%A _Michel Lagneau_, Mar 26 2011
%E Offset corrected by _Arkadiusz Wesolowski_, Jun 28 2011
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