OFFSET
0,2
COMMENTS
The denominators are given in A130414.
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.
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).
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.
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
EXAMPLE
Rationals r(n): 1, 19/18, 47/45, 1321/1260, 989/945, 21779/20790, 141481/135135, ...
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
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Jun 01 2007
STATUS
approved