A054387 Numerators of coefficients of 1/2^(2n+1) in Newton's series for Pi.

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

Author

Eric W. Weisstein

Keywords

• sign

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

Examples

			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) - ...)
		

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

Crossrefs

Cf. A054388.

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).