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A158933 Decimal expansion of Sum_{n>=1} ((-1)^(n+1))/F(n) where F(n) is the n-th Fibonacci number A000045(n). 3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

LINKS

Table of n, a(n) for n=0..94.

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 W. Weisstein, MathWorld: Reciprocal Fibonacci Constant.

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

EXAMPLE

0.2891446485706715831123055096157291669...

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 *)

PROG

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

CROSSREFS

Cf. A000045, A001622, A079586.

Cf. A153386, A153387.

Sequence in context: A016641 A155748 A152748 * A200502 A316138 A011062

Adjacent sequences:  A158930 A158931 A158932 * A158934 A158935 A158936

KEYWORD

nonn,cons

AUTHOR

Michel Lagneau, Mar 26 2011

EXTENSIONS

Offset corrected by Arkadiusz Wesolowski, Jun 28 2011

STATUS

approved

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Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)