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A344041 Decimal expansion of Sum_{k>=1} F(k)/(k*2^k), where F(k) is the k-th Fibonacci number (A000045). 0
8, 6, 0, 8, 1, 7, 8, 8, 1, 9, 2, 8, 0, 0, 8, 0, 7, 7, 7, 7, 8, 8, 6, 6, 4, 6, 5, 9, 0, 1, 2, 1, 0, 8, 5, 0, 8, 4, 9, 1, 4, 1, 3, 6, 5, 0, 8, 0, 5, 7, 9, 3, 0, 9, 5, 1, 4, 0, 1, 2, 2, 0, 7, 9, 8, 5, 1, 2, 2, 4, 3, 0, 9, 2, 2, 2, 6, 3, 9, 2, 2, 7, 2, 2, 9, 8, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This constant is a transcendental number (Adhikari et al., 2001).

A similar series is Sum_{k>=1} F(k)/2^k = 2.

The corresponding series with Lucas numbers (A000032) is Sum_{k>=1} L(k)/(k*2^k) = 2*log(2) (A016627).

In general, for m>=2, Sum_{k>=1} F(k)/(k*m^k) = log(1 - 2*sqrt(5)/(1 + sqrt(5) - 2*m)) / sqrt(5) and Sum_{k>=1} L(k)/(k*m^k) = log(m^2 / (m^2 - m - 1)). - Vaclav Kotesovec, May 08 2021

LINKS

Table of n, a(n) for n=0..86.

S. D. Adhikari, N. Saradha, T. N. Shorey and R. Tijdeman, Transcendental infinite sums, Indagationes Mathematicae, Vol. 12, No. 1 (2001), pp. 1-14.

István Mező, Several Generating Functions for Second-Order Recurrence Sequences, Journal of Integer Sequences, Vol. 12 (2009), Article 09.3.7.

FORMULA

Equals Sum_{k>=0} (-1)^k/A002457(k).

Equals 4*log(phi)/sqrt(5) = 4*arcsinh(1/2)/sqrt(5) = arccosh(7/2)/sqrt(5) = 4*A002390/A002163.

Equals Integral_{x>=2} 1/(x^2 - x - 1) dx.

EXAMPLE

0.86081788192800807777886646590121085084914136508057...

MATHEMATICA

RealDigits[Sum[Fibonacci[n]/n/2^n, {n, 1, Infinity}], 10, 100][[1]]

PROG

(PARI) suminf(k=1, fibonacci(k)/(k*2^k)) \\ Michel Marcus, May 07 2021

CROSSREFS

Cf. A000032, A000045, A002163, A002390, A002457, A036289.

Sequence in context: A069855 A156551 A074738 * A240805 A010115 A268438

Adjacent sequences:  A344038 A344039 A344040 * A344042 A344043 A344044

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, May 07 2021

STATUS

approved

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Last modified July 26 20:28 EDT 2021. Contains 346295 sequences. (Running on oeis4.)