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A338305 Decimal expansion of Sum_{k>=0} 1/F(2^k+1), where F(k) is the k-th Fibonacci number (A000045).

%I A338305 #11 Jan 14 2023 13:32:40
%S A338305 1,7,3,0,0,3,8,2,2,2,5,0,4,2,4,3,2,4,2,3,0,4,1,2,3,5,6,6,4,9,6,8,9,9,
%T A338305 0,1,0,3,4,7,9,5,5,0,0,4,8,1,0,3,0,9,4,1,5,5,5,6,7,0,8,7,7,7,5,5,8,0,
%U A338305 1,1,6,0,8,0,9,7,2,2,6,0,4,5,3,7,3,7,3
%N A338305 Decimal expansion of Sum_{k>=0} 1/F(2^k+1), where F(k) is the k-th Fibonacci number (A000045).
%C A338305 Erdős and Graham (1980) asked whether this constant is irrational or transcendental.
%C A338305 Badea (1987) proved that it is irrational.
%C A338305 Becker and Töpper (1994) proved that it is transcendental.
%C A338305 Note that a similar sum, Sum_{k>=0} 1/F(2^k) = (7-sqrt(5))/2 (A079585), is quadratic rational in Q(sqrt(5)).
%H A338305 Catalin Badea, <a href="https://doi.org/10.1017/S0017089500006868">The irrationality of certain infinite series</a>, Glasgow Mathematical Journal, Vol. 29, No. 2 (1987), pp. 221-228.
%H A338305 Paul-Georg Becker and Thomas Töpper, <a href="https://doi.org/10.1002/mana.19941680102">Transcendency results for sums of reciprocals of linear recurrences</a>, Mathematische Nachrichten, Vol. 168, No. 1 (1994), pp. 5-17.
%H A338305 Paul Erdős and Ronald L. Graham, <a href="http://www.math.ucsd.edu/~fan/ron/papers/80_11_number_theory.pdf">Old and new problems and results in combinatorial number theory</a>, L'enseignement Mathématique, Université de Genève, 1980, p. 64-65.
%H A338305 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A338305 Equals Sum_{k>=0} 1/A192222(k).
%e A338305 1.73003822250424324230412356649689901034795500481030...
%t A338305 RealDigits[Sum[1/Fibonacci[2^n + 1], {n, 0, 10}], 10, 100][[1]]
%o A338305 (PARI) suminf(k=0, 1/fibonacci(2^k+1)) \\ _Michel Marcus_, Oct 21 2020
%Y A338305 Cf. A000045, A079585, A192222, A338304.
%K A338305 nonn,cons
%O A338305 1,2
%A A338305 _Amiram Eldar_, Oct 21 2020