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.

A383082 The number of distinct straightedge-and-compass constructions that can be made with a total of n lines and circles.

Original entry on oeis.org

1, 3, 3, 16, 205, 5886, 542983
Offset: 0

Views

Author

Peter Kagey, Apr 15 2025

Keywords

Comments

A straightedge-and-compass construction starts with 2 points marked on the plane, traditionally (0,0) and (1,0). One can use a straightedge to draw a line between two marked points or a compass to draw a circle centered at a marked point through another marked point.
New points occur at the intersections of lines or circles with lines or circles.
Constructions are considered distinct if the location of the lines, circles, and points is different. In particular, a translation of a construction is considered distinct. - Peter Kagey, May 07 2025

Examples

			For n = 0, the a(0) = 1 diagram is the one consisting of two points.
For n = 1, there are a(1) = 3 possible constructions:
1) a line between (0,0), and (1,0),
2) a circle of radius 1 centered at (0,0),
3) a circle of radius 1 centered at (1,0).
For n = 2, there are a(2) = 3 possible constructions:
1) a line between (0,0), and (1,0) and a circle of radius 1 centered at (0,0), which marks the point (-1,0);
2) a line between (0,0), and (1,0) and a circle of radius 1 centered at (1,0), which marks the point (2,0);
3) two circles, both of radius 1, centered at (0,0) and (1,0), which marks the points (1/2,sqrt(3/4)) and (1/2,-sqrt(3/4)).
For n=3, see the a(3)=16 diagrams in the link.
		

Crossrefs

A383083 The number of distinct straightedge-and-compass constructions that can be made with no lines and n circles.

Original entry on oeis.org

1, 2, 1, 4, 44, 1084, 91192
Offset: 0

Views

Author

Peter Kagey, Apr 16 2025

Keywords

Comments

A straightedge-and-compass construction starts with 2 points marked on the plane, traditionally (0,0) and (1,0). In the constructions counted by this sequence, only the compass is used. Circles can be drawn at any marked point through any other marked point, and new points are marked where circles intersect.

Crossrefs

A383084 The number of points in the Euclidean plane that can be determined via a straightedge-and-compass construction using n or fewer lines and circles.

Original entry on oeis.org

2, 2, 6, 14, 147, 5743, 900487
Offset: 0

Views

Author

Peter Kagey, Apr 16 2025

Keywords

Examples

			For both n = 0 and n = 1, the a(0) = a(1) = 2 points are the initial points: (0,0) and (1,0).
For n = 2, the a(2) = 6 points are the two above points along with (-1,0), (2,0), (1/2,-sqrt(3)/2), and (1/2,sqrt(3)/2):
  (-1,0): formed by intersecting the line between (0,0) and (1,0) with the circle of radius 1 centered at (0,0).
  (2,0): formed by intersecting the line between (0,0) and (1,0) with the circle of radius 1 centered at (1,0).
  (1/2,-sqrt(3)/2), and (1/2,sqrt(3)/2): formed by intersecting the circle of radius 1 centered at (0,0) with the circle of radius 1 centered at (1,0).
		

Crossrefs

A383273 Triangle read by rows: T(n,k) is the number of ruler-and-compass constructions consisting of n-k lines and k circles with 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 0, 0, 12, 4, 0, 0, 45, 116, 44, 0, 0, 232, 1565, 3005, 1084, 0, 0, 1627, 34114, 166556, 249494, 91192, 0, 0, 21547
Offset: 0

Views

Author

Peter Kagey, Apr 21 2025

Keywords

Comments

Row sums are given by A383082.

Examples

			n\k | 0 1   2    3    4    5
----+--------------------------
  0 | 1
  1 | 1 2
  2 | 0 2    1
  3 | 0 0   12     4
  4 | 0 0   45   116     44
  5 | 0 0  232  1565   3005   1084
  6 | 0 0 1627 34114 166556 249494 91192
T(2,1) = 2 because there are two ruler-and-compass constructions with a line and a circle:
  A circle centered at (0,0) through (1,0) and a line through (0,0),(1,0).
  A circle centered at (1,0) through (0,0) and a line through (0,0),(1,0).
T(2,2) = 1 because there is one ruler-and-compass construction with two circles:
  A circle centered at (0,0) through (1,0) and a circle centered at (1,0) through (0,0).
		

Crossrefs

Formula

T(n,n) = A383083(n).
Showing 1-4 of 4 results.