

A053002


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


4



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
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OFFSET

1,3


COMMENTS

On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1}(1/sqrt(1t^4)).
M(a,b) is the limit of the arithmeticgeometric 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
OEIS Wiki, Gauss's constant


EXAMPLE

0.83462684167407318628142973...


MATHEMATICA

ContinuedFraction[1/ArithmeticGeometricMean[1, Sqrt[2]] , 100] (* JeanFrançois 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, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009


CROSSREFS

Cf. A014549.
Sequence in context: A156148 A224867 A156824 * A053003 A167202 A204914
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


STATUS

approved



