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

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

Offset

1,2

Comments

Erdős and Graham (1980) asked whether this constant is irrational or transcendental.

Badea (1987) proved that it is irrational.

Becker and Töpper (1994) proved that it is transcendental.

Note that a similar sum, Sum_{k>=0} 1/F(2^k) = (7-sqrt(5))/2 (A079585), is quadratic rational in Q(sqrt(5)).

Links

Table of n, a(n) for n=1..87.

Catalin Badea, The irrationality of certain infinite series, Glasgow Mathematical Journal, Vol. 29, No. 2 (1987), pp. 221-228.

Paul-Georg Becker and Thomas Töpper, Transcendency results for sums of reciprocals of linear recurrences, Mathematische Nachrichten, Vol. 168, No. 1 (1994), pp. 5-17.

Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980, p. 64-65.

Index entries for transcendental numbers

Formula

Equals Sum_{k>=0} 1/A192222(k).

Example

1.73003822250424324230412356649689901034795500481030...

Mathematica

RealDigits[Sum[1/Fibonacci[2^n + 1], {n, 0, 10}], 10, 100][[1]]

Prog

(PARI) suminf(k=0, 1/fibonacci(2^k+1)) \\ Michel Marcus, Oct 21 2020

Crossrefs

Cf. A000045, A079585, A192222, A338304.

Sequence in context: A100983 A350621 A103237 * A239120 A369134 A021582

Adjacent sequences: A338302 A338303 A338304 * A338306 A338307 A338308

Keyword

nonn,cons

Author

Amiram Eldar, Oct 21 2020

Status

approved