A121502 -id:A121502 - cp's OEIS frontend

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

Wolfdieter Lang, Aug 16 2006

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.

			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].

Wolfdieter Lang, Sequence of decreasing relative errors and more.

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

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).

A121506 Minimal polygon values appearing in a certain polygon problem leading to an approximation of Pi.

Original entry on oeis.org

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, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66

Offset: 3

Wolfdieter Lang, Aug 16 2006

Analog of A121500 with n and m roles interchanged.
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).
This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with a 0.15% relative error. See the Popper reference under A121503.

			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).
m=7, a(7)=9=n: F(9,7) leads to error E(9,7)= 0.003122 (rounded).
This is larger than E(8,6), therefore the m value 7 does not appear in A121502.
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 hexagon lead to a larger relative error.

Cf. A121502 (values for m for which relative errors E(n, m) decrease).

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.

cp's OEIS Frontend

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

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