A142969 Numerators of approximants of a continued fraction for 4/Pi - 1 = (4 - Pi)/Pi.

1, 2, 29, 52, 887, 8066, 11069, 143128, 3485197, 2792362, 78773861, 326941444, 1166735057, 28815727078, 1038855637093, 902109848368, 1031041592023, 33635927876926, 37917122954701, 1387635433109516, 66513954553071413, 59972573887236398, 3113073102662686381

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 Padé 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 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.

Example

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

Crossrefs

Sequence in context: A366692 A059799 A367796 * A281546 A115448 A276169

Adjacent sequences: A142966 A142967 A142968 * A142970 A142971 A142972

Keyword

nonn,easy,frac,cofr

Author

Wolfdieter Lang, Sep 15 2008

Status

approved