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%I #16 Aug 29 2019 16:27:42
%S 1,2,61,44,989,6346,51197,36056,4127401,2057402,189721879,236723324,
%T 1422382919,20600649518,10227626700773,638723926928,1278290544991,
%U 23635180313246,94585786464329,969106771716436,83372817133541471
%N Numerator of partial sums of a series for 3*(Pi-3).
%C Denominators are given in A130412.
%C The rationals (in lowest terms) r(n):=3*sum(((-1)^(j+1))/(j*(j+1)*(2*j+1)),j=1..n) have the limit 3*(Pi-3), approximately 0.424777962, for n->infinity.
%C These partial sums result from those for the more familiar series s(n):=sum(((-1)^(j+1))/(2*j*(2*j+1)*(2*j+2)),j=1..n) with limit (Pi-3)/4 which is approximately 0.0353981635. r(n)= 12*s(n). This series is attributed to K. G. Nilakantha, see, e.g., the R. Roy reference. eq.(13).
%C The sum r(n)/3 gives the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/6+...Proof with Euler's 1748 conversion of continued fractions into series. The denominators q(n)=A001879 of the n-th approximant of this continued fraction is used. The author (WL) reconsidered this entry after an e-mail from R. Rosenthal Jul 16 2008 pointing out the Pi-3 continued fraction.
%H W. Lang, <a href="/A130411/a130411.txt">Rationals and limit.</a>
%H Ranjan Roy, <a href="http://www.jstor.org/stable/2690896">The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha</a>, Math. Magazine 63 (1990), 291-306. Reprinted in: Pi: A Source Book, eds. L. Berggren, et al., Springer, New York, 1997, pp. 92-107.
%F a(n) = numerator(r(n)) with the rationals r(n) given above.
%e Rationals r(n), n>=1: [1/2, 2/5, 61/140, 44/105, 989/2310, 6346/15015, 51197/120120, ...].
%e Rationals s(n)=r(n)/12, n>=1: [1/24, 1/30, 61/1680, 11/315, 989/27720, 3173/90090, 51197/1441440, ...].
%K nonn,frac,easy
%O 1,2
%A _Wolfdieter Lang_, Jun 01 2007, Sep 09 2008, Oct 06 2008
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