%I #12 Sep 20 2023 05:50:54

%S 3,5,6,8,11,14,15,17,18,21,31,38,48,65,82,89,99,106,123,181,222,280,

%T 379,478,519,577,618,717

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

%C 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 Wolfdieter Lang, <a href="/A121501/a121501.txt">Sequence of decreasing relative errors and more</a>.

%F 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 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 (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 Cf. A121502 (corresponding A121500(a(k)) numbers).

%K nonn,more

%O 1,1

%A _Wolfdieter Lang_, Aug 16 2006