A121506 Minimal polygon values appearing in a certain polygon problem leading to an approximation of Pi.

3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66

Offset: 3

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• easy

Comments

Analog of A121500 with n and m roles interchanged.

For a regular m-gon circumscribed around a unit circle (area Pi) the arithmetic mean of the areas of this m-gon with a regular inscribed n-gon is nearest to Pi for n=a(m).

This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with a 0.15% relative error. See the Popper reference under A121503.

Examples

			m=15, a(15)=21=n: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error E(21,15)= 0.0000147(rounded).
m=7, a(7)=9=n: F(9,7) leads to error E(9,7)= 0.003122 (rounded).
This is larger than E(8,6), therefore the m value 7 does not appear in A121502.
m=6, a(6)=8=n: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error E(8,6)= 0.001487 (rounded). All other inscribed n-gons with circumscribed hexagon lead to a larger relative error.
		

Crossrefs

Cf. A121502 (values for m for which relative errors E(n, m) decrease).

Formula

a(m)=min(abs(F(n,m)),n=3..infinity), m>=3 (checked for n=3..3+500), with F(nm):= ((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 an regular n-gon circumscribing the unit circle. E(n,m) = (F(n,m)-pi)/pi is the relative error.