A053003 Continued fraction for M(1,sqrt(2)).

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, 1

Offset: 0

Author

N. J. A. Sloane, Feb 21 2000

Keywords

• nonn
• cofr
• nice
• easy

Comments

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

			1.19814023473559220743992249228...
		

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

Crossrefs

Cf. A014549, A053002 without the leading term, A053004 (decimal expansion).

Programs

Mathematica

ContinuedFraction[ArithmeticGeometricMean[1,Sqrt[2]],100] (* Harvey P. Dale, Feb 26 2012 *)

PARI

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(agm(1, sqrt(2))); for (n=1, 20000, write("b053003.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