A121501 - OEIS

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

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..28.

Wolfdieter Lang, Sequence of decreasing relative errors and more.

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).

EXAMPLE

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].

CROSSREFS

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

Sequence in context: A327206 A212439 A160734 * A327218 A062832 A089085

Adjacent sequences: A121498 A121499 A121500 * A121502 A121503 A121504

KEYWORD

nonn,more

AUTHOR

Wolfdieter Lang, Aug 16 2006

STATUS

approved