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Search: id:A053002
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| A053002 |
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Continued fraction for 1 / M(1,sqrt(2)) (Gauss's constant). |
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+0
4
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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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).
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REFERENCES
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J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,20000
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
G. Xiao, Contfrac
Index entries for continued fractions for constants
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EXAMPLE
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0.83462684167407318628142973...
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PROGRAM
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(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]
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CROSSREFS
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Cf. A014549.
Sequence in context: A002030 A156148 A156824 this_sequence A053003 A167202 A165204
Adjacent sequences: A052999 A053000 A053001 this_sequence A053003 A053004 A053005
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KEYWORD
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nonn,cofr,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Feb 21 2000
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 22 2000
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