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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). 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 (list; constant; graph; refs; listen; history; text; internal format)
 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 Cf. A000045, A079585, A192222, A338304. Sequence in context: A038268 A100983 A103237 * A239120 A021582 A098459 Adjacent sequences:  A338302 A338303 A338304 * A338306 A338307 A338308 KEYWORD nonn,cons AUTHOR Amiram Eldar, Oct 21 2020 STATUS approved

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Last modified September 20 16:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)