OFFSET
0,1
COMMENTS
Since the perimeter equals n*sin(180ยบ/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).
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
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Robert G. Wilson v, Jun 27 2015
STATUS
approved