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A210395
Decimal expansion of continued fraction with quotients equal to Fermat numbers.
0
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.
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
KEYWORD
nonn,cons
AUTHOR
Algirdas Javtokas, Mar 21 2012
EXTENSIONS
Offset changed by Bruno Berselli, May 14 2012
STATUS
approved