A385330 The point numbers encountered by a rotating marker following the process described in the Comments.

1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 1, 3, 1, 2, 4, 1, 3, 1, 2, 2, 2, 3, 1, 4, 3, 1, 2, 5, 1, 4, 1, 2, 3, 1, 3, 2, 2, 4, 2, 3, 1, 5, 4, 3, 6, 1, 1, 2, 5, 2, 1, 4, 3, 1, 2, 3, 4, 1, 3, 2, 2, 5, 4, 2, 6, 3, 1, 5, 7, 1, 4, 3, 6, 1, 1, 2, 2, 5, 3, 2, 1, 4, 4, 3, 1, 2, 5

Offset

1,3

Comments

A marker moves clockwise in a unit circle around the origin. On revolution k, the regular star with k points is laid down with points labeled (k,1), (k,2), ..., (k,k) at regular angles. As the marker rotates we output both the newly laid down points and any existing points the marker sweeps over. The first point of the k-th star is placed at an angle bisecting the last point of the previous star and the next point already laid down.

If a new point happens to coincide exactly with an existing point, then the existing label is replaced with the appropriate new label.

The process begins with the degenerate 1-star, laying down the label (1,1) at 0 degrees. Next, the 2-star is laid down, starting with label (2,1) at 180 degrees (180 degrees being considered the bisector of 0 and 360 degrees), then overwriting (1,1) with (2,2) at 0 degrees. Next, the 3-star is laid down, starting at 90 degrees (the bisecting angle of 0 and 180 degrees) with (3,1), followed by (3,2) at 90+360/3 degrees, and so on. See the illustration in the Links and the Example.

Informally, this sequence is reminiscent of a radar display sweeping round and round gradually identifying more and more targets.

Conjecture: The only time a label is replaced is laying down the second point of the 2-star which overwrites the first (and only point) from the 1-star.

Links

Table of n, a(n) for n=1..87.

Sean A. Irvine, Java program (github)

Tamas Sandor Nagy, Illustration for the construction of the sequence.

Example

The sequence can be written as an irregular triangle, read by rows, with row n corresponding to laying down the n-star. The first few rows are (star-number, point-number):
   (1,1);
   (2,1), (2,2)*;
   (3,1), (2,1), (3,2), (3,3);
   (4,1), (2,2), (4,2), (3,1), (4,3), (2,1), (3,2), (4,4);
   (5,1), (3,3), (4,1), (2,2), (5,2), (4,2), (5,3), (3,1), (5,4), (4,3), (2,1), (3,2), (5,5);
   ...
   * The label (2,2) overwrites the (1,1) label (the only known occurrence of an overwrite).

Crossrefs

Cf. A385615.

Sequence in context: A063726 A290267 A240750 * A181118 A179009 A112757

Adjacent sequences: A385327 A385328 A385329 * A385331 A385332 A385333

Keyword

nonn,tabf

Author

Tamas Sandor Nagy, Jun 25 2025

Extensions

More terms from Sean A. Irvine, Jun 30 2025

Status

approved