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COMMENTS
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The rationals r(n)=2*n*e(n-1)/e(n), where e(n)=A000111(n), approximate Pi as n -> oo. - M. F. Hasler, Apr 03 2013
Numerators are given in A132049.
See the Delahaye reference and a link by W. Lang given in A132049.
From Paul Curtz, Mar 17 2013: (Start)
Apply the Akiyama-Tanigawa transform (or algorithm) to A046978(n+2)/A016116(n+1):
1, 1/2, 0, -1/4, -1/4, -1/8, 0, 1/16, 1/16;
1/2, 1, 3/4, 0, -5/8, -3/4, -7/16, 0; = Balmer0(n)
-1/2, 1/2, 9/4, 5/2, 5/8, -15/8, -49/16;
-1, -7/2, -3/4, 15/2, 25/2, 57/8;
5/2, -11/2, -99/4, -20, 215/8;
8, 77/2, -57/4, -375/2;
-61/2, 211/2, 2079/4;
-136, -1657/2;
1385/2;
The first column is PIEULER(n) = 1, 1/2, -1/2, -1, 5/2, 8, -61/2, -136, 1385/2,... = c(n)/d(n). Abs c(n+1)=1,1,1,5,8,61,... =a(n) with offset=1.
For numerators of Balmer0(n) see A076109, A000265 and A061037(n-1) (End).
Other completely unrelated rational approximations of Pi are given by A063674/A063673 and other references there. - M. F. Hasler, Apr 03 2013
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EXAMPLE
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Rationals r(n): [2, 4, 3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521,...].
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