This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A130411 #16 Aug 29 2019 16:27:42 %S A130411 1,2,61,44,989,6346,51197,36056,4127401,2057402,189721879,236723324, %T A130411 1422382919,20600649518,10227626700773,638723926928,1278290544991, %U A130411 23635180313246,94585786464329,969106771716436,83372817133541471 %N A130411 Numerator of partial sums of a series for 3*(Pi-3). %C A130411 Denominators are given in A130412. %C A130411 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 A130411 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 A130411 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 A130411 W. Lang, <a href="/A130411/a130411.txt">Rationals and limit.</a> %H A130411 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 A130411 a(n) = numerator(r(n)) with the rationals r(n) given above. %e A130411 Rationals r(n), n>=1: [1/2, 2/5, 61/140, 44/105, 989/2310, 6346/15015, 51197/120120, ...]. %e A130411 Rationals s(n)=r(n)/12, n>=1: [1/24, 1/30, 61/1680, 11/315, 989/27720, 3173/90090, 51197/1441440, ...]. %K A130411 nonn,frac,easy %O A130411 1,2 %A A130411 _Wolfdieter Lang_, Jun 01 2007, Sep 09 2008, Oct 06 2008