A259442 a(n) is the least number of sides of a regular circumscribed k-gon whose perimeter yields Pi to within 1/10^n.

4, 11, 33, 102, 322, 1017, 3215, 10167, 32149, 101664, 321488, 1016633, 3214876, 10166330, 32148757, 101663296, 321487567, 1016632951, 3214875668, 10166329505, 32148756680, 101663295049, 321487566791, 1016632950485, 3214875667907, 10166329504841, 32148756679070, 101663295048410

Offset: 0

Author

Robert G. Wilson v, Jun 27 2015

Keywords

• easy
• nonn

Comments

Since the perimeter equals n*tan(180º/n), increasing n to greater values will yield a more accurate value of Pi.

Lim n -> inf., a(n+1)/a(n) = sqrt(10). This implies that a(n+2) ~ 10*a(n).

Lim n -> inf., a(2n) = 10^n*sqrt(Pi^3/3) and a(2n+1) = 10^n*sqrt(Pi^3/30).

Lim n -> inf., a(n)/A259441(n) = sqrt(2).

Examples

			a(0) # 3 since the perimeter of the circumscribed triangle is sqrt(27) which equals approximately 5.196152... which exceeds Pi by more than 1;
a(0) = 4 since the perimeter of the circumscribed square is 4 and this is within 1 of the true value of Pi;
a(1) = 11 since the perimeter of the circumscribed 11-gon which equals approximately 3.229891... which is within 0.1 of the true value of Pi;
a(2) = 33 since the perimeter of the circumscribed 33-gon which equals approximately 3.151117... which is within 0.01 of the true value of Pi; etc.
		

References

• William H. Beyer, Ed., CRC Standard Mathematical Tables, 27th Ed., IV - Geometry, Mensuration Formulas, p. 122, Boca Raton 1984.
• Daniel Zwillinger, Editor-in-Chief, 31st Ed., CRC Standard Mathematical Tables and Formulae, 4.5.3 Geometry - Regular Polygons, p. 324, Boca Raton, 2003.
• Jan Gullberg, Mathematics: From the Birth of Numbers, 13.3 Solving Triangles, p. 479, W. W. Norton & Co., NY, 1997.
• Catherine A. Gorini, Ph.D., The Facts on File Geometry Handbook, Charts & Tables, p. 262, Checkmark Books, NY, 2005.

Links

• mathematicsonline, How to Calculate Pi using Archimedes' Method

Crossrefs

Cf. A000796, A244644, A259441.

Programs

Mathematica

f[n_] := Block[{k = Floor[ Sqrt[ 10]*f[n - 1]] - 6}, While[Pi + 10^-n < k*Tan[Pi/k], k++]; k]; f[-1] = 3; Array[f, 28, 0]