cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A121502 Companion sequence for A121501 used in a unit circle area approximation problem.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 11, 12, 13, 15, 22, 27, 34, 46, 58, 63, 70, 75, 87, 128, 157, 198
Offset: 1

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Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

The pairs (n(k),m(k)):=(A121501(k),a(k)), k>=1, lead to a strictly decreasing sequence of relative errors E(n(k),m(k)) in the circle area approximation problem described in A121500. For each k=1,2,... the unit circle has inscribed n(k)-gon and circumscribed m(k)-gon.
For the sequence of relative errors E(n(k),m(k)), k=1..20 see the W. Lang link in A121501.

Examples

			(n(k),m(k)) pairs (A121501(k),a(k)), for k=1..7: [3, 3], [5, 4], [6, 5], [8, 6], [11, 8], [14, 10], [15, 11].

Formula

a(k) = A121500(A121501(k)), k>=1.

A121500 Minimal polygon values for a certain polygon problem leading to an approximation of Pi.

Original entry on oeis.org

3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42
Offset: 3

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

For a regular n-gon inscribed in a unit circle (area Pi), the arithmetic mean of the areas of this n-gon with a regular circumscribed m-gon is nearest to Pi for m=a(n).
This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with 0.15% relative error. See the Popper reference under A121503.

Examples

			n=8, a(8)=6: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error 0.001487 (rounded). All other circumscribed m-gons with inscribed octagon lead to a larger relative error.
n=21, a(21)=15: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error 0.0000147 (rounded).

Crossrefs

Cf. A121501 (positions n where relative errors decrease).

Formula

a(n) = min(abs(E(n,m)),m >= 3), n>=3 (checked for m=3..3+500), with E(n,m):= ((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 a regular n-gon circumscribing the unit circle.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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