A051158 Decimal expansion of Sum_{n >= 0} 1/(2^2^n+1).

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

Offset: 0

Author

Robert Lozyniak (11(AT)onna.com)

Keywords

• nonn
• cons

Examples

			0.59606317211782167942...
		

Links

• 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.

Crossrefs

A048649 + A051158 = 2.

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

Cf. A000215, A000120, A007814.

Programs

Mathematica

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

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

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