%I A121500
%S A121500 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,
%T A121500 21,22,23,23,24,25,26,26,27,28,28,29,30,30,31,32,33,33,34,35,35,36,37,
%U A121500 38,38,39,40,40,41,42,42
%N A121500 Minimal polygon values for a certain polygon problem leading to an approximation 
               of pi.
%C A121500 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 circumsribed m-gon 
               is nearest to pi for m=a(n).
%C A121500 This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) 
               which approximates Pi with 1.5 permille relative error. See the Popper 
               reference under A121503.
%F A121500 a(n)=min(abs(E(n,m)),m=3..infinity), 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 A121500 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 
               8-gon lead to a larger relative error.
%e A121500 n=21, a(21)=15: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 
               digits) leads to a relative error 0.0000147 (rounded).
%Y A121500 Cf. A121501 (positions n where relative errors decrease).
%Y A121500 Sequence in context: A054760 A079107 A023963 this_sequence A113455 A054637 
               A120172
%Y A121500 Adjacent sequences: A121497 A121498 A121499 this_sequence A121501 A121502 
               A121503
%K A121500 nonn,easy
%O A121500 3,1
%A A121500 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 
               2006