%I #17 May 22 2022 17:42:43

%S 1,2,29,52,887,8066,11069,143128,3485197,2792362,78773861,326941444,

%T 1166735057,28815727078,1038855637093,902109848368,1031041592023,

%U 33635927876926,37917122954701,1387635433109516,66513954553071413,59972573887236398,3113073102662686381

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

%C Denominators are A007509(n), n >= 1.

%C This results from William Brouncker's continued fraction for 4/Pi without the leading 1.

%C William Brouncker's result appears in John Wallis's "Arithmetica infinitorum" from 1655.

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

%H Wolfdieter Lang, <a href="/A142969/a142969.txt">Rationals and more.</a>

%F 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+...

%F 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.

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

%K nonn,easy,frac,cofr

%O 1,2

%A _Wolfdieter Lang_, Sep 15 2008