A259441 a(n) is the least number of sides of a regular inscribed k-gon whose perimeter yields Pi to within 1/10^n.

3, 8, 23, 72, 228, 719, 2274, 7189, 22733, 71887, 227327, 718869, 2273261, 7188681, 22732604, 71886806, 227326039, 718868054, 2273260386, 7188680533, 22732603855, 71886805327, 227326038545, 718868053265, 2273260385449, 7188680532650, 22732603854487, 71886805326500

Offset: 0

Author

Robert G. Wilson v, Jun 27 2015

Keywords

• base
• easy
• nonn

Comments

Since the perimeter equals n*sin(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^3/6) and a(2n+1) = 10^n*sqrt(Pi^3/60).

Lim n -> inf., A259442(n)/a(n) = sqrt(2).

Examples

			a(0) = 3 since the perimeter of an inscribed triangle is sqrt(27)/2 which equals approximately 2.598076... and this is within 1.0 of Pi's true value;
a(1) = 8 since the perimeter of an inscribed octagon is 4*sqrt(2 - sqrt(2)) which equals approximately 3.061467... and this is within 0.1 of Pi's true value;
a(2) = 23 since the perimeter of an inscribed 23-gon is approximately 3.131832... and this in within 0.01 of Pi's true value; 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, A259442.

Programs

Mathematica

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