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

Author

Wolfdieter Lang, Sep 15 2008

Keywords

• nonn
• easy
• frac
• cofr

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.

Examples

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

References

• C. Brezinski, History of Continued Fractions and Padé approximants, Springer, 1991, ch. 3.

Links

• 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.