login
A338305
Decimal expansion of Sum_{k>=0} 1/F(2^k+1), where F(k) is the k-th Fibonacci number (A000045).
2
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
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.
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
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 21 2020
STATUS
approved