A344041 Decimal expansion of Sum_{k>=1} F(k)/(k*2^k), where F(k) is the k-th Fibonacci number (A000045).

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

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.

Index entries for transcendental numbers

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