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 A130413 #17 Aug 30 2019 03:54:08
%S A130413 1,19,47,1321,989,21779,141481,1132277,801821,91424611,45706007,
%T A130413 4205393539,5256312899,31539920369,457304942543,226832956041173,
%U A130413 14176557010703,28353956712541,524535004412921,2098185082863029
%N A130413 Numerators of partial sums for a series for Pi/3.
%C A130413 The denominators are given in A130414.
%C A130413 The rationals r(n) = 1 + (4/3)*Sum_{j=1..n} (-1)^(j+1)/((2*j+1)*((2*j+1)^2-1)), n >= 0, have the limit lim_{n->infinity} r(n) = Pi/3, approximately 1.047197551.
%C A130413 This series is obtained from the one for Pi/4 (attributed to Nilakantha) obtained by multiplication with 3/4. See the R. Roy link eq.(13).
%H A130413 Robert Israel, <a href="/A130413/b130413.txt">Table of n, a(n) for n = 0..1000</a>
%H A130413 W. Lang, <a href="/A130413/a130413.txt">Rationals and limit</a>
%H A130413 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.
%F A130413 a(n) = numerator(r(n)), n >= 0, with r(n) defined above.
%F A130413 G.f. for r(n): 4*arctan(sqrt(x))/(3*sqrt(x)*(1-x)) - log(x+1)/(3*x). - _Robert Israel_, Jul 27 2015
%e A130413 Rationals r(n): 1, 19/18, 47/45, 1321/1260, 989/945, 21779/20790, 141481/135135, ...
%p A130413 f:= n -> numer(1+ (4/3)*add(((-1)^(j+1))/((2*j+1)*((2*j+1)^2-1)),j=1..n)):
%p A130413 map(f, [$0..20]); # _Robert Israel_, Jul 27 2015
%Y A130413 Cf. A130411/A130412 (partial sums for a series of 3*(Pi-3)).
%K A130413 nonn,frac,easy
%O A130413 0,2
%A A130413 _Wolfdieter Lang_, Jun 01 2007