login
A130413
Numerators of partial sums for a series for Pi/3.
2
1, 19, 47, 1321, 989, 21779, 141481, 1132277, 801821, 91424611, 45706007, 4205393539, 5256312899, 31539920369, 457304942543, 226832956041173, 14176557010703, 28353956712541, 524535004412921, 2098185082863029
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).
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
Cf. A130411/A130412 (partial sums for a series of 3*(Pi-3)).
Sequence in context: A141973 A165672 A136686 * A378051 A156376 A063306
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Jun 01 2007
STATUS
approved