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.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

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Author

Wolfdieter Lang, Aug 16 2006

Keywords

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

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).

A121505 Hit triangle for unit circle area (Pi) approximation problem described in A121500.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 3

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Record for n=3,4,... only those (n, A121500(n)) pairs which have relative error E(n, A121500(n)) smaller than all errors with smaller n. This produces the table a(n,m).
The unit circle area is approximated by the arithmetic mean of the areas of an inscribed regular n-gon and a circumscribed regular m-gon.
For each row n>=3 the minimal relative error E(n,m):= ((Fin(n) + Fout(m))/2-Pi)/ Pi) appears for m= A121500(n).
The same hit triangle is obtained when one considers the minimal relative errors for the columns m>=3 and collects the sequence with decreasing errors, starting with m=3.

Examples

			[1], [0,0], [0,1,0], [0, 0, 1, 0], [0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0],...

Links

Formula

a(n,m) = 1 if m = A121500(n) and E(n,m) < min(E(k,A121500(k)), k=3..n-1), n>=4. a(3,3) = 1, else a(n,m) = 0.

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

Views

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.

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

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Formula

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.
Showing 1-4 of 4 results.

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