A121506 - OEIS

%I #6 Sep 16 2013 00:41:42

%S 3,5,6,8,9,11,12,14,15,17,18,20,21,22,24,25,27,28,30,31,32,34,35,37,

%T 38,39,41,42,44,45,47,48,49,51,52,54,55,56,58,59,61,62,64,65,66,66,66,

%U 66,66,66,66,66,66,66,66,66

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

%C Analog of A121500 with n and m roles interchanged.

%C 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).

%C 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.

%F 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.

%e 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).

%e m=7, a(7)=9=n: F(9,7) leads to error E(9,7)= 0.003122 (rounded).

%e This is larger than E(8,6), therefore the m value 7 does not appear in A121502.

%e 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.

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

%K nonn,easy

%O 3,1

%A _Wolfdieter Lang_, Aug 16 2006