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

3, 5, 6, 8, 11, 14, 15, 17, 18, 21, 31, 38, 48, 65, 82, 89, 99, 106, 123, 181, 222, 280, 379, 478, 519, 577, 618, 717

Offset: 1

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• more

Comments

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.

Examples

			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.
(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].
		

Links

• Wolfdieter Lang, Sequence of decreasing relative errors and more.

Crossrefs

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

Formula

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).