A121500 Minimal polygon values for a certain polygon problem leading to an approximation of Pi.

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

Offset

3,1

Comments

For a regular n-gon inscribed in a unit circle (area Pi), the arithmetic mean of the areas of this n-gon with a regular circumscribed m-gon 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))/2-Pi)/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 n-gon inscribed in the unit circle. Fout(n) is the area of a regular n-gon 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 m-gons 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