

A142969


Numerators of approximants of a continued fraction for 4/Pi  1 = (4  Pi)/Pi.


3



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

1,2


COMMENTS

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.


REFERENCES

C. Brezinski, History of Continued Fractions and PadÃ© approximants, Springer, 1991, ch. 3.


LINKS

Table of n, a(n) for n=1..23.
Wolfdieter Lang, Rationals and more.


FORMULA

a(n) = numerator(C(n)) with C(n) the nth 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*k1))^2/(q(k)*q(k1)), 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.


EXAMPLE

Approximants a(n)/A007509(n): 1/2, 2/13, 29/76, 52/263, 887/2578, 8066/36979, ...


CROSSREFS

Sequence in context: A330895 A105893 A059799 * A281546 A115448 A276169
Adjacent sequences: A142966 A142967 A142968 * A142970 A142971 A142972


KEYWORD

nonn,easy,frac,cofr


AUTHOR

Wolfdieter Lang, Sep 15 2008


STATUS

approved



