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Search: id:A142969
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| A142969 |
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Numerators of approximants of a continued fraction for 4/Pi-1 = (4-Pi)/Pi. |
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+0
2
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| 1, 2, 29, 52, 887, 8066, 11069, 143128, 3485197, 2792362, 78773861, 326941444, 1166735057, 28815727078, 1038855637093, 902109848368, 1031041592023, 33635927876926, 37917122954701, 1387635433109516, 66513954553071413, 59972573887236398, 3113073102662686381
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OFFSET
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1,2
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COMMENT
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Denominators are A007509(n), n>=1.
This results from William Brouncker's continued fraction for 4/Pi without the leading 1.
William Brouncker's result appears in John Wallis' "Arithmetica infinitorum" from 1655.
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REFERENCES
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C. Brezinski, History of Continued Fractions and Pad\'e approximants, Springer, 1991, ch. 3.
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LINKS
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W. Lang, Rationals and more.
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FORMULA
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a(n)=numerator(C(n)) with C(n) the n-th approximant to the continued fraction (1^2)(2+(3^2)/(2+(5^2)/(2+...
C(n)=sum(((-1)^(k+1))*(product(2*k-1,j=1..k)^2)/(q(k)*q(k-1)),k=1..n), with q(n):=A024199(n+1). Proof with Euler's conversion of continued fractions to alternating series. For this conversion see e.g. the Brezinski reference, p. 98.
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EXAMPLE
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Approximants a(n)/A007509(n): [1/2], [2/13], [29/76], [52/263], [887/2578], [8066/36979], ...
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CROSSREFS
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Sequence in context: A078329 A105893 A059799 this_sequence A115448 A107161 A041097
Adjacent sequences: A142966 A142967 A142968 this_sequence A142970 A142971 A142972
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KEYWORD
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nonn,easy,frac,cofr
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 15 2008
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