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%I #13 Mar 02 2015 02:11:17
%S 3,1,9,7,6,7,4,9,4,4,5,8,7,6,5,5,9,3,6,4,1,1,6,2,8,9,0,2,1,7,5,2,4,4,
%T 8,0,2,1,2,7,8,3,5,2,5,4,1,4,9,1,5,7,1,9,2,5,7,5,1,4,9,3,1,6,9,9,2,9,
%U 2,8,9,3,2,1,5,9,9,2,6,8,0,0,7,9,9,5,5,7,8,7,2,6
%N Decimal expansion of continued fraction with quotients equal to Fermat numbers.
%D A. Ya. Khinchin, Continued Fractions, Dover Publications, 1997.
%D M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer, 2011.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Continued_fraction">Continued fraction</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>
%F a(0).a(1)a(2)a(3)a(4)a(5)... = F_0 + 1/(F_1 + 1/(F_2 + 1/(F_3 + 1/(F_4 + ...)))) = [F_0,F_1,F_2,F_3,F_4,...] where a(0).a(1)a(2)a(3)a(4)... is a decimal representation of the continued fraction [F_0,F_1,F_2,F_3,F_4,...] where F_0, F_1,... are Fermat numbers.
%e 3.19767494... = 3 + 1/(5 + 1/(17 + 1/(257 + 1/(65537 + ...))))
%t FromContinuedFraction[{3,5,17,257,65537,4294967297,18446744073709551617}] (* for better precision, enter next Fermat numbers *)
%o (PARI) s=3;forstep(n=log(default(realprecision)*log(10)\log(2))\log(2), 1,-1,s=1/(2.^(2^n)+s+1));s \\ _Charles R Greathouse IV_, Mar 21 2012
%Y Cf. A000215.
%K nonn,cons
%O 1,1
%A _Algirdas Javtokas_, Mar 21 2012
%E Offset changed by _Bruno Berselli_, May 14 2012
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