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A121501 Positions n of A121500 where the minimal relative error associated with the polygon problem described there decreases.

%I A121501 #12 Sep 20 2023 05:50:54
%S A121501 3,5,6,8,11,14,15,17,18,21,31,38,48,65,82,89,99,106,123,181,222,280,
%T A121501 379,478,519,577,618,717
%N A121501 Positions n of A121500 where the minimal relative error associated with the polygon problem described there decreases.
%C A121501 The minimal relative errors for the unit circle area approximation by the arithmetic mean of areas of an inscribed regular n-gon and a circumscribed regular A121500(n)-gon decrease (strictly) for these n=a(k) values. This results from a minimization, first within row n and then along the rows n of the matrix E(n,m) defined below.
%H A121501 Wolfdieter Lang, <a href="/A121501/a121501.txt">Sequence of decreasing relative errors and more</a>.
%F A121501 a(k) is such that E(a(k),A121500(a(k)) < min(E(n,A121500(n)),n=3..a(k)-1), k>=2, a(1):=3, with the relative error E(n,m):= abs(F(n,m)-Pi))/Pi and F(n,m):= (Fin(n)+Fout(m))/2, where Fin(n):=(n/2)*sin(2*Pi/ n) and Fout(m):= m*tan(Pi/m).
%e A121501 For k=4, a(4)=8, m:= A121500(8)= 6. The relative error associated with F(n=8,m=6) is the smallest among those with values n=3,..,8.
%e A121501 (n,m) pairs (a(k),A121500(a(k)), for k=1..7: [3, 3], [5, 4], [6, 5], [8, 6], [11, 8], [14, 10], [15, 11].
%Y A121501 Cf. A121502 (corresponding A121500(a(k)) numbers).
%K A121501 nonn,more
%O A121501 1,1
%A A121501 _Wolfdieter Lang_, Aug 16 2006