%I A121506
%S A121506 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,
%T A121506 39,41,42,44,45,47,48,49,51,52,54,55,56,58,59,61,62,64,65,66,66,66,66,
%U A121506 66,66,66,66,66,66,66,66
%N A121506 Minimal polygon values appearing in a certain polygon problem leading 
               to an approximation of pi.
%C A121506 Analog of A121500 with n and m roles interchanged.
%C A121506 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 A121506 This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) 
               which approximates Pi with a 1.5 permille relative error. See the 
               Popper reference under A121503.
%F A121506 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 A121506 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 A121506 m=7, a(7)=9=n: F(9,7) leads to error E(9,7)= 0.003122 (rounded).
%e A121506 This is larger than E(8,6), therefore the m value 7 does not appear in 
               A121502.
%e A121506 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 
               6-gon lead to a larger relative error.
%Y A121506 Cf. A121502 (values for m for which relative errors E(n, m) decrease).
%Y A121506 Sequence in context: A133561 A095117 A089585 this_sequence A114119 A101358 
               A047446
%Y A121506 Adjacent sequences: A121503 A121504 A121505 this_sequence A121507 A121508 
               A121509
%K A121506 nonn,easy
%O A121506 3,1
%A A121506 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 
               2006