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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).

%I A344041 #18 Jun 07 2024 10:09:26
%S A344041 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,
%T A344041 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,
%U A344041 2,4,3,0,9,2,2,2,6,3,9,2,2,7,2,2,9,8,0
%N A344041 Decimal expansion of Sum_{k>=1} F(k)/(k*2^k), where F(k) is the k-th Fibonacci number (A000045).
%C A344041 This constant is a transcendental number (Adhikari et al., 2001).
%C A344041 A similar series is Sum_{k>=1} F(k)/2^k = 2.
%C A344041 The corresponding series with Lucas numbers (A000032) is Sum_{k>=1} L(k)/(k*2^k) = 2*log(2) (A016627).
%C A344041 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
%H A344041 S. D. Adhikari, N. Saradha, T. N. Shorey and R. Tijdeman, <a href="https://doi.org/10.1016/S0019-3577(01)80001-X">Transcendental infinite sums</a>, Indagationes Mathematicae, Vol. 12, No. 1 (2001), pp. 1-14.
%H A344041 István Mező, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mezo/mezo5.html">Several Generating Functions for Second-Order Recurrence Sequences</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.3.7.
%H A344041 Renzo Sprugnoli, <a href="https://www.emis.de/journals/INTEGERS/papers/g27/g27.Abstract.html">Sums of reciprocals of the central binomial coefficients</a>, INTEGERS 6 (2006) #A27.
%H A344041 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A344041 Equals Sum_{k>=0} (-1)^k/A002457(k).
%F A344041 Equals 4*log(phi)/sqrt(5) = 4*arcsinh(1/2)/sqrt(5) = arccosh(7/2)/sqrt(5) = 4*A002390/A002163.
%F A344041 Equals Integral_{x>=2} 1/(x^2 - x - 1) dx.
%e A344041 0.86081788192800807777886646590121085084914136508057...
%t A344041 RealDigits[Sum[Fibonacci[n]/n/2^n, {n, 1, Infinity}], 10, 100][[1]]
%o A344041 (PARI) suminf(k=1, fibonacci(k)/(k*2^k)) \\ _Michel Marcus_, May 07 2021
%Y A344041 Cf. A000032, A000045, A002163, A002390, A002457, A036289.
%K A344041 nonn,cons
%O A344041 0,1
%A A344041 _Amiram Eldar_, May 07 2021