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

1,2

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).

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 = 1..20000

Eric Weisstein's World of Mathematics, Gauss's Constant.

G. Xiao, Contfrac

Index entries for continued fractions for constants

Example

1.19814023473559220743992249228...

Mathematica

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

Prog

(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(agm(1, sqrt(2))); for (n=1, 20000, write("b053003.txt", n, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

Crossrefs

Cf. A014549, A053002 without the leading term, A053004.

Sequence in context: A224867 A156824 A053002 * A346035 A167202 A204914

Adjacent sequences: A053000 A053001 A053002 * A053004 A053005 A053006

Keyword

nonn,cofr,nice,easy

Author

N. J. A. Sloane, Feb 21 2000

Extensions

More terms from James A. Sellers, Feb 22 2000

Status

approved