A158933 Decimal expansion of Sum_{n>=1} ((-1)^(n+1))/F(n) where F(n) is the n-th Fibonacci number A000045(n).

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, 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, 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

Offset: 0

Author

Michel Lagneau, Mar 26 2011

Keywords

• nonn
• cons

Comments

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

Examples

			0.2891446485706715831123055096157291669...
		

Links

• Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
• Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
• Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.

Crossrefs

Cf. A000045, A001622, A079586.

Cf. A153386, A153387, A324007.

Programs

Maple

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):

Mathematica

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 *)

PARI

-sumalt(n=1,(-1)^n/fibonacci(n)) \\ Charles R Greathouse IV, Oct 03 2016

Formula

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

Equals A153387 - A153386. - Joerg Arndt, Oct 04 2020

Equals 1 - A324007. - Amiram Eldar, Feb 09 2023

Extensions

Offset corrected by Arkadiusz Wesolowski, Jun 28 2011