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A051158 Decimal expansion of Sum_{n >= 0} 1/(2^2^n+1). 5
5, 9, 6, 0, 6, 3, 1, 7, 2, 1, 1, 7, 8, 2, 1, 6, 7, 9, 4, 2, 3, 7, 9, 3, 9, 2, 5, 8, 6, 2, 7, 9, 0, 6, 4, 5, 4, 6, 2, 3, 6, 1, 2, 3, 8, 4, 7, 8, 1, 0, 9, 9, 3, 2, 6, 2, 1, 4, 4, 2, 4, 5, 9, 9, 6, 0, 9, 1, 0, 8, 9, 9, 7, 7, 4, 8, 8, 6, 0, 8, 8, 8, 9, 9, 3, 6, 1, 9, 1, 8, 4, 6, 4, 6, 4, 4, 0, 7, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Table of n, a(n) for n=0..98.

Joerg Arndt, Matters Computational (The Fxtbook), section 38.7, p.740 (gives method for divisionless computation corresponding to PARI/GP code below).

S. Audinarayana Moorthy, Problem E2455, The American Mathematical Monthly, Vol. 81, No. 1 (1974), p. 85, solution, ibid., Vol. 82, No. 2 (1975), pp. 173-174.

Michael Coons, On the rational approximation of the sum of the reciprocals of the Fermat numbers, Raman. J., Vol. 28 (2013), pp. 39-65.

Michael Coons, Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers, arXiv:1511.08147 [math.NT], 2015.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 247.

Solomon W. Golomb, On the sum of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math., Vol. 15 (1963), pp. 475-478.

FORMULA

Equals (1/2) * Sum_{k>=1} A000120(k)/2^k (S. Audinarayana Moorthy, 1974). - Amiram Eldar, May 15 2020

Equals 1 - Sum_{n>=1} A007814(n)/2^n = 2/3 - Sum_{n>=1} A007814(n)/4^n = 3/5 - Sum_{n>=1} A007814(n)/16^n. - Amiram Eldar, Nov 06 2020

EXAMPLE

0.59606317211782167942...

MATHEMATICA

RealDigits[Sum[1/(2^2^n + 1), {n, 0, 10}], 10, 111][[1]] (* Robert G. Wilson v, Jul 03 2014 *)

PROG

(PARI) /* divisionless routine from fxtbook */

s2(y, N=7)=

{ local(in, y2, A); /* as powerseries correct to order = 2^N-1 */

    in = 1; /* 1+y+y^2+y^3+...+y^(2^k-1) */

    A = y; for(k=2, N, in *= (1+y); y *= y; A += y*(in + A); );

    return( A ); }

a=0.5*s2(0.5) /* computation of the constant 0.596063172117821... */

/* Joerg Arndt, Apr 15 2010 */

(PARI) suminf(n=0, 1/(2^2^n+1)) \\ Michel Marcus, May 15 2020

CROSSREFS

A048649 + A051158 = 2.

Terms in continued fraction: A159243. - Enrique Pérez Herrero, Nov 17 2009

Cf. A000120, A007814.

Sequence in context: A057821 A133742 A134879 * A117605 A303983 A073003

Adjacent sequences:  A051155 A051156 A051157 * A051159 A051160 A051161

KEYWORD

nonn,cons

AUTHOR

Robert Lozyniak (11(AT)onna.com)

STATUS

approved

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Last modified July 1 10:54 EDT 2022. Contains 354972 sequences. (Running on oeis4.)