OFFSET
0,1
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.
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
KEYWORD
nonn,cons
AUTHOR
Robert Lozyniak (11(AT)onna.com)
STATUS
approved