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%I #12 Mar 28 2015 22:37:40
%S 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,
%T 21,21,22,23,23,24,25,26,26,27,28,28,29,30,30,31,32,33,33,34,35,35,36,
%U 37,38,38,39,40,40,41,42,42
%N Minimal polygon values for a certain polygon problem leading to an approximation of Pi.
%C 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).
%C 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.
%F 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.
%e 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.
%e n=21, a(21)=15: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error 0.0000147 (rounded).
%Y Cf. A121501 (positions n where relative errors decrease).
%K nonn,easy
%O 3,1
%A _Wolfdieter Lang_, Aug 16 2006
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