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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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3
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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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COMMENTS
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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's "Arithmetica infinitorum" from 1655.
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REFERENCES
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C. Brezinski, History of Continued Fractions and Padé approximants, Springer, 1991, ch. 3.
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LINKS
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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_{k=1..n} (-1)^(k+1)*(Product_{j=1..k} (2*k-1))^2/(q(k)*q(k-1)), 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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KEYWORD
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nonn,easy,frac,cofr
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AUTHOR
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STATUS
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approved
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