

A130411


Numerator of partial sums of a series for 3*(Pi3).


4



1, 2, 61, 44, 989, 6346, 51197, 36056, 4127401, 2057402, 189721879, 236723324, 1422382919, 20600649518, 10227626700773, 638723926928, 1278290544991, 23635180313246, 94585786464329, 969106771716436, 83372817133541471
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OFFSET

1,2


COMMENTS

Denominators are given in A130412.
The rationals (in lowest terms) r(n):=3*sum(((1)^(j+1))/(j*(j+1)*(2*j+1)),j=1..n) have the limit 3*(Pi3), approximately 0.424777962, for n>infinity.
These partial sums result from those for the more familiar series s(n):=sum(((1)^(j+1))/(2*j*(2*j+1)*(2*j+2)),j=1..n) with limit (Pi3)/4 which is approximately 0.0353981635. r(n)= 12*s(n). This series is attributed to K. G. Nilakantha, see, e.g., the R. Roy reference. eq.(13).
The sum r(n)/3 gives the nth approximant to the continued fraction 1^2/(6+3^2/(6+5^2/6+...Proof with Euler's 1748 conversion of continued fractions into series. The denominators q(n)=A001879 of the nth approximant of this continued fraction is used. The author (WL) reconsidered this entry after an email from R. Rosenthal Jul 16 2008 pointing out the Pi3 continued fraction.


LINKS

Table of n, a(n) for n=1..21.
W. Lang, Rationals and limit.
Ranjan Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Math. Magazine 63 (1990), 291306. Reprinted in: Pi: A Source Book, eds. L. Berggren, et al., Springer, New York, 1997, pp. 92107.


FORMULA

a(n) = numerator(r(n)) with the rationals r(n) given above.


EXAMPLE

Rationals r(n), n>=1: [1/2, 2/5, 61/140, 44/105, 989/2310, 6346/15015, 51197/120120, ...].
Rationals s(n)=r(n)/12, n>=1: [1/24, 1/30, 61/1680, 11/315, 989/27720, 3173/90090, 51197/1441440, ...].


CROSSREFS

Sequence in context: A078491 A182856 A101896 * A262079 A222009 A041449
Adjacent sequences: A130408 A130409 A130410 * A130412 A130413 A130414


KEYWORD

nonn,frac,easy


AUTHOR

Wolfdieter Lang, Jun 01 2007, Sep 09 2008, Oct 06 2008


STATUS

approved



