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

%I A259441 #15 Jul 15 2015 17:49:32
%S A259441 3,8,23,72,228,719,2274,7189,22733,71887,227327,718869,2273261,
%T A259441 7188681,22732604,71886806,227326039,718868054,2273260386,7188680533,
%U A259441 22732603855,71886805327,227326038545,718868053265,2273260385449,7188680532650,22732603854487,71886805326500
%N A259441 a(n) is the least number of sides of a regular inscribed k-gon whose perimeter yields Pi to within 1/10^n.
%C A259441 Since the perimeter equals n*sin(180º/n), increasing n to greater values will yield a more accurate value of Pi.
%C A259441 Lim n -> inf., a(n+1)/a(n) = sqrt(10). This implies that a(n+2) ~ 10*a(n).
%C A259441 Lim n -> inf., a(2n) = 10^n*sqrt(Pi^3/6) and a(2n+1) = 10^n*sqrt(Pi^3/60).
%C A259441 Lim n -> inf., A259442(n)/a(n) = sqrt(2).
%D A259441 William H. Beyer, Ed., CRC Standard Mathematical Tables, 27th Ed., IV - Geometry, Mensuration Formulas, p. 122, Boca Raton 1984.
%D A259441 Daniel Zwillinger, Editor-in-Chief, 31st Ed., CRC Standard Mathematical Tables and Formulae, 4.5.3 Geometry - Regular Polygons, p. 324, Boca Raton, 2003.
%D A259441 Jan Gullberg, Mathematics: From the Birth of Numbers, 13.3 Solving Triangles, p. 479, W. W. Norton & Co., NY, 1997.
%D A259441 Catherine A. Gorini, Ph.D., The Facts on File Geometry Handbook, Charts & Tables, p. 262, Checkmark Books, NY, 2005.
%H A259441 mathematicsonline, <a href="https://www.youtube.com/watch?v=DLZMZ-CT7YU/"> How to Calculate Pi using Archimedes' Method</a>
%e A259441 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;
%e A259441 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;
%e A259441 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.
%t A259441 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]
%Y A259441 Cf. A000796, A244644, A259442.
%K A259441 base,easy,nonn
%O A259441 0,1
%A A259441 _Robert G. Wilson v_, Jun 27 2015