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
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
KEYWORD
nonn,easy,frac,cofr
AUTHOR
Wolfdieter Lang, Sep 15 2008
STATUS
approved