%I #43 Oct 24 2024 17:00:32

%S 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,

%T 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,

%U 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

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

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 38.7, p.740 (gives method for divisionless computation corresponding to PARI/GP code below).

%H S. Audinarayana Moorthy, <a href="https://www.jstor.org/stable/2318925">Problem E2455</a>, The American Mathematical Monthly, Vol. 81, No. 1 (1974), p. 85, <a href="https://www.jstor.org/stable/2319669">solution</a>, ibid., Vol. 82, No. 2 (1975), pp. 173-174.

%H Michael Coons, <a href="http://dx.doi.org/10.1007/s11139-012-9410-x">On the rational approximation of the sum of the reciprocals of the Fermat numbers</a>, Raman. J., Vol. 28 (2013), pp. 39-65.

%H Michael Coons, <a href="http://arxiv.org/abs/1511.08147">Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers</a>, arXiv:1511.08147 [math.NT], 2015.

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 247.

%H Solomon W. Golomb, <a href="http://dx.doi.org/10.4153/CJM-1963-051-0">On the sum of the reciprocals of the Fermat numbers and related irrationalities</a>, Canad. J. Math., Vol. 15 (1963), pp. 475-478.

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

%F 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

%e 0.59606317211782167942...

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

%o (PARI) /* divisionless routine from fxtbook */

%o s2(y, N=7)=

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

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

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

%o return( A ); }

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

%o /* _Joerg Arndt_, Apr 15 2010 */

%o (PARI) suminf(n=0, 1/(2^2^n+1)) \\ _Michel Marcus_, May 15 2020

%Y A048649 + A051158 = 2.

%Y Terms in continued fraction: A159243. - _Enrique Pérez Herrero_, Nov 17 2009

%Y Cf. A000215, A000120, A007814.

%K nonn,cons

%O 0,1

%A Robert Lozyniak (11(AT)onna.com)