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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
OFFSET
0,2
COMMENTS
According to Beckmann, Newton undertook his Pi calculations in Woolsthorpe during the plague years of 1665-6. Actually, Newton was calculating something else, and Pi appeared only as an incidental fringe benefit in the calculation. Twenty-two 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. 140-143.
LINKS
A. Sofo, Pi and some other constants, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005.
Eric Weisstein's World of Mathematics, Pi Formulas
FORMULA
Pi = 3*sqrt(3)/4 + 24*(1/12 - sum(n >= 2, (2*n-2)!/((n-1)!^2*(2*n-3)*(2*n+1)*2^(4*n-2)))) (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 * A380241 A199958 A112734
KEYWORD
sign
STATUS
approved