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

Author

Amiram Eldar, Oct 21 2020

Keywords

• nonn
• cons

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)).

Examples

			1.73003822250424324230412356649689901034795500481030...
		

Links

• 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

Crossrefs

Cf. A000045, A079585, A192222, A338304.

Programs

Mathematica

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

PARI

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

Formula

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