

A121500


Minimal polygon values for a certain polygon problem leading to an approximation of Pi.


4



3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42
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OFFSET

3,1


COMMENTS

For a regular ngon inscribed in a unit circle (area Pi), the arithmetic mean of the areas of this ngon with a regular circumscribed mgon is nearest to Pi for m=a(n).
This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with 0.15% relative error. See the Popper reference under A121503.


LINKS

Table of n, a(n) for n=3..60.


FORMULA

a(n) = min(abs(E(n,m)),m >= 3), n>=3 (checked for m=3..3+500), with E(n,m):= ((Fin(n)+Fout(m))/2Pi)/Pi), where Fin(n):=(n/2)*sin(2*Pi/n) and Fout(m):= m*tan(Pi/m). Fin(n) is the area of the regular ngon inscribed in the unit circle. Fout(n) is the area of a regular ngon circumscribing the unit circle.


EXAMPLE

n=8, a(8)=6: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error 0.001487 (rounded). All other circumscribed mgons with inscribed octagon lead to a larger relative error.
n=21, a(21)=15: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error 0.0000147 (rounded).


CROSSREFS

Cf. A121501 (positions n where relative errors decrease).
Sequence in context: A257923 A288178 A023963 * A182230 A113455 A054637
Adjacent sequences: A121497 A121498 A121499 * A121501 A121502 A121503


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Aug 16 2006


STATUS

approved



