A053002 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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).

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

EXAMPLE

0.83462684167407318628142973...

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A014549 (decimal expansion).

Sequence in context: A156148 A224867 A156824 * A053003 A373163 A346035

Adjacent sequences: A052999 A053000 A053001 * A053003 A053004 A053005

KEYWORD

nonn,cofr,nice,easy

AUTHOR

N. J. A. Sloane, Feb 21 2000

EXTENSIONS

More terms from James A. Sellers, Feb 22 2000

Offset changed by Andrew Howroyd, Aug 03 2024

STATUS

approved