A153386 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n).

1, 5, 3, 5, 3, 7, 0, 5, 0, 8, 8, 3, 6, 2, 5, 2, 9, 8, 5, 0, 2, 9, 8, 5, 2, 8, 9, 6, 6, 5, 1, 5, 9, 9, 0, 0, 6, 3, 6, 7, 0, 1, 1, 5, 9, 1, 0, 7, 1, 1, 3, 8, 5, 6, 3, 2, 3, 5, 2, 6, 3, 6, 6, 5, 1, 3, 1, 0, 4, 7, 2, 7, 8, 6, 2, 8, 9, 0, 9, 4, 1, 6, 0, 1, 6, 5, 0, 2, 3, 1, 6, 6, 3, 6, 9, 6, 9, 3, 3, 6, 5, 3, 2, 7, 9

Offset

1,2

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.14.1, p. 358.

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See exercise 114 at p. 270.

Links

Table of n, a(n) for n=1..105.

Richard André-Jeannin, Problem B-745, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 31, No. 3 (1993), p. 277; Fun with Unit Fractions, Solution to Problem B-745 by Paul S. Bruckman, ibid., Vol. 33, No. 1 (1995), p. 86.

A. F. Horadam, Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences, The Fibonacci Quarterly, Vol. 26, No. 2 (1988), pp. 98-114.

Formula

Equals sqrt(5) * (L((3-sqrt(5))/2) - L((7-3*sqrt(5))/2)), where L(x) = Sum_{k>=1} x^k/(1-x^k) (Horadam, 1988, equation (4.6)). - Amiram Eldar, Oct 04 2020

From Gleb Koloskov, Sep 04 2021: (Start)

Equals 1/2 + (sqrt(5)/log(phi))*(log(5)/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(exp(Pi*x/log(phi))-1)) dx), where phi = (1+sqrt(5))/2 = A001622.

Equals 1/2 + (A002163/A002390)*(A016628/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(A001113^(A000796*x/A002390)-1)) dx). (End)

Equals 1 + Sum_{n>=1} 1/A065563(2*n-1) (André-Jeannin, 1993). - Amiram Eldar, Jan 15 2022

From Peter Bala, Aug 17 2022: (Start)

Equals 5/3 - 3*Sum_{n >= 1} 1/(F(2*n)*F(2*n+2)*F(2*n+4)), where F(n) = Fibonacci(n).

Conjecture: Equals 151/96 - 6*Sum_{n >= 1} 1/(F(2*n)*F(2*n+4)*F(2*n+6)). (End)

Equals A360928 * sqrt(5). - Kevin Ryde, Feb 27 2023

Example

1.535370508836252985029852896651599006367...

Mathematica

rd[k_] := rd[k] = RealDigits[ N[ Sum[ 1/Fibonacci[2*n], {n, 1, 2^k}], 105]][[1]]; rd[k = 4]; While[ rd[k] != rd[k - 1], k++]; rd[k] (* Jean-François Alcover, Oct 29 2012 *)
RealDigits[Sqrt[5] * (Log[5] + 2*QPolyGamma[0, 1, 1/GoldenRatio^4] - 4*QPolyGamma[0, 1, 1/GoldenRatio^2]) / (8*ArcCsch[2]), 10, 105][[1]] (* Vaclav Kotesovec, Feb 26 2023 *)

Prog

(PARI) sumpos(n=1, 1/fibonacci(2*n)) \\ Michel Marcus, Sep 04 2021

Crossrefs

Cf. A000045, A001906 (Fibonacci(2*n)), A065563.

Cf. A000796, A001113, A002163, A002390, A016628, A079586, A153387, A256178, A265288, A360928.

Sequence in context: A270915 A319461 A201762 * A112920 A109364 A122277

Adjacent sequences: A153383 A153384 A153385 * A153387 A153388 A153389

Keyword

nonn,cons

Author

Eric W. Weisstein, Dec 25 2008

Status

approved