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A051158
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Decimal expansion of Sum_{n >= 0} 1/(2^2^n+1).
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5
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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
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OFFSET
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0,1
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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0.59606317211782167942...
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MATHEMATICA
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RealDigits[Sum[1/(2^2^n + 1), {n, 0, 10}], 10, 111][[1]] (* Robert G. Wilson v, Jul 03 2014 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,cons
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AUTHOR
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Robert Lozyniak (11(AT)onna.com)
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STATUS
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approved
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