|
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,3
|
|
|
COMMENTS
| On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral(1/sqrt(1-t^4),t=0..1).
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
OEIS Wiki, Gauss's constant
|
|
|
EXAMPLE
| 0.83462684167407318628142973...
|
|
|
MATHEMATICA
| ContinuedFraction[1/ArithmeticGeometricMean[1, Sqrt[2]] , 100] (* From 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, " ", x[n])); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 20 2009]
|
|
|
CROSSREFS
| Cf. A014549.
Sequence in context: A202860 A156148 A156824 * A053003 A167202 A204914
Adjacent sequences: A052999 A053000 A053001 * A053003 A053004 A053005
|
|
|
KEYWORD
| nonn,cofr,nice,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Feb 21 2000
|
|
|
EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 22 2000
|
| |
|
|
