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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; text; internal format)
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(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
Eric Weisstein's World of Mathematics, Gauss's Constant
G. Xiao, Contfrac
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, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009
CROSSREFS
Cf. A014549.
Sequence in context: A156148 A224867 A156824 * A053003 A346035 A167202
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

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)