A053002 Continued fraction for 1 / M(1,sqrt(2)) (Gauss's constant).

0, 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, 2, 2, 8, 3, 1, 2, 2, 1, 10, 2, 2, 2, 3, 3, 1, 7, 1, 8, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 2, 17, 1, 4, 31, 2, 2, 5, 30, 1, 8, 2

Offset: 0

Author

N. J. A. Sloane, Feb 21 2000

Keywords

• nonn
• cofr
• nice
• easy

Comments

On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1}(1/sqrt(1-t^4)).

M(a,b) is the limit of the arithmetic-geometric mean iteration applied repeatedly starting with a and b: a_0=a, b_0=b, a_{n+1}=(a_n+b_n)/2, b_{n+1}=sqrt(a_n*b_n).

Examples

			0.83462684167407318628142973...
		

References

• J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5.
• J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..19999
• Eric Weisstein's World of Mathematics, Gauss's Constant
• G. Xiao, Contfrac
• Index entries for continued fractions for constants
• OEIS Wiki, Gauss's constant

Crossrefs

Cf. A014549 (decimal expansion).

Programs

Mathematica

ContinuedFraction[1/ArithmeticGeometricMean[1, Sqrt[2]] , 100]  (* Jean-François Alcover, Apr 18 2011 *)

PARI

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(1/agm(1, sqrt(2))); for (n=1, 20000, write("b053002.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

Extensions

More terms from James Sellers, Feb 22 2000

Offset changed by Andrew Howroyd, Aug 03 2024