A210395 Decimal expansion of continued fraction with quotients equal to Fermat numbers.

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, 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, 2, 8, 9, 3, 2, 1, 5, 9, 9, 2, 6, 8, 0, 0, 7, 9, 9, 5, 5, 7, 8, 7, 2, 6

Offset

1,1

References

A. Ya. Khinchin, Continued Fractions, Dover Publications, 1997.

M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer, 2011.

Links

Table of n, a(n) for n=1..92.

Wikipedia, Continued fraction

Wikipedia, Fermat number

Formula

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.

Example

3.19767494... = 3 + 1/(5 + 1/(17 + 1/(257 + 1/(65537 + ...))))

Mathematica

FromContinuedFraction[{3, 5, 17, 257, 65537, 4294967297, 18446744073709551617}] (* for better precision, enter next Fermat numbers *)

Prog

(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

Crossrefs

Cf. A000215.

Sequence in context: A143495 A327997 A245789 * A019770 A136320 A201840

Adjacent sequences: A210392 A210393 A210394 * A210396 A210397 A210398

Keyword

nonn,cons

Author

Algirdas Javtokas, Mar 21 2012

Extensions

Offset changed by Bruno Berselli, May 14 2012

Status

approved