A130413 - OEIS

%I #17 Aug 30 2019 03:54:08

%S 1,19,47,1321,989,21779,141481,1132277,801821,91424611,45706007,

%T 4205393539,5256312899,31539920369,457304942543,226832956041173,

%U 14176557010703,28353956712541,524535004412921,2098185082863029

%N Numerators of partial sums for a series for Pi/3.

%C The denominators are given in A130414.

%C 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 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 Robert Israel, <a href="/A130413/b130413.txt">Table of n, a(n) for n = 0..1000</a>

%H W. Lang, <a href="/A130413/a130413.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.

%F a(n) = numerator(r(n)), n >= 0, with r(n) defined above.

%F 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 Rationals r(n): 1, 19/18, 47/45, 1321/1260, 989/945, 21779/20790, 141481/135135, ...

%p f:= n -> numer(1+ (4/3)*add(((-1)^(j+1))/((2*j+1)*((2*j+1)^2-1)),j=1..n)):

%p map(f, [$0..20]); # _Robert Israel_, Jul 27 2015

%Y Cf. A130411/A130412 (partial sums for a series of 3*(Pi-3)).

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 01 2007