A130413 Numerators of partial sums for a series for Pi/3.
1, 19, 47, 1321, 989, 21779, 141481, 1132277, 801821, 91424611, 45706007, 4205393539, 5256312899, 31539920369, 457304942543, 226832956041173, 14176557010703, 28353956712541, 524535004412921, 2098185082863029
Offset: 0
Examples
Rationals r(n): 1, 19/18, 47/45, 1321/1260, 989/945, 21779/20790, 141481/135135, ...
Links
- Robert Israel, Table of n, a(n) for n = 0..1000
- W. Lang, Rationals and limit
- Ranjan Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Math. Magazine 63 (1990), 291-306.
Programs
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Maple
f:= n -> numer(1+ (4/3)*add(((-1)^(j+1))/((2*j+1)*((2*j+1)^2-1)),j=1..n)): map(f, [$0..20]); # Robert Israel, Jul 27 2015
Formula
a(n) = numerator(r(n)), n >= 0, with r(n) defined above.
G.f. for r(n): 4*arctan(sqrt(x))/(3*sqrt(x)*(1-x)) - log(x+1)/(3*x). - Robert Israel, Jul 27 2015
Comments