

A054387


Numerators of coefficients of 1/2^(2n+1) in Newton's series for Pi.


2



0, 2, 1, 1, 1, 5, 7, 7, 33, 429, 715, 2431, 4199, 29393, 52003, 185725, 111435, 1938969, 17678835, 21607465, 119409675, 883631595, 109402007, 6116566755, 11435320455, 57176602275, 322476036831, 1215486600363, 2295919134019
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OFFSET

0,2


COMMENTS

According to Beckmann, Newton undertook his Pi calculations in Woolsthorpe during the plague years of 16656. Actually, Newton was calculating something else, and Pi appeared only as an incidental fringe benefit in the calculation. Twentytwo terms were sufficient to give him 16 decimal places (the last was incorrect because of the inevitable error in rounding off).  Johannes W. Meijer, Feb 23 2013


REFERENCES

Petr Beckmann, A history of Pi, 1974, pp. 140143.


LINKS

Table of n, a(n) for n=0..28.
A. Sofo, Pi and some other constants, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005.
Eric W. Weisstein, MathWorld: Pi Formulas


FORMULA

Pi = 3*sqrt(3)/4 + 24*(1/12  sum(n >= 2, (2*n2)!/((n1)!^2*(2*n3)*(2*n+1)*2^(4*n2)))) (Newton).


EXAMPLE

Pi = 3*sqrt(3)/4 + 24*(0/(1*2) + 2/(3*2^3)  1/(5*2^5)  1/(28*2^7)  1/(72*2^9)  ...)


CROSSREFS

Cf. A054388.
Sequence in context: A219585 A292464 A090628 * A199958 A112734 A260685
Adjacent sequences: A054384 A054385 A054386 * A054388 A054389 A054390


KEYWORD

sign


AUTHOR

Eric W. Weisstein


STATUS

approved



