

A259441


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


1



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
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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, EditorinChief, 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

Table of n, a(n) for n=0..27.
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 23gon 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

Cf. A000796, A244644, A259442.
Sequence in context: A148776 A127385 A152880 * A176605 A080410 A230952
Adjacent sequences: A259438 A259439 A259440 * A259442 A259443 A259444


KEYWORD

base,easy,nonn


AUTHOR

Robert G. Wilson v, Jun 27 2015


STATUS

approved



