cp's OEIS Frontend

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

%I A385330 #32 Jul 09 2025 05:31:40
%S A385330 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,
%T A385330 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,
%U A385330 7,1,4,3,6,1,1,2,2,5,3,2,1,4,4,3,1,2,5
%N A385330 The point numbers encountered by a rotating marker following the process described in the Comments.
%C A385330 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.
%C A385330 If a new point happens to coincide exactly with an existing point, then the existing label is replaced with the appropriate new label.
%C A385330 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.
%C A385330 Informally, this sequence is reminiscent of a radar display sweeping round and round gradually identifying more and more targets.
%C A385330 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.
%H A385330 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a385/A385330.java">Java program</a> (github)
%H A385330 Tamas Sandor Nagy, <a href="/A385330/a385330.pdf">Illustration for the construction of the sequence.</a>
%e A385330 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):
%e A385330    (1,1);
%e A385330    (2,1), (2,2)*;
%e A385330    (3,1), (2,1), (3,2), (3,3);
%e A385330    (4,1), (2,2), (4,2), (3,1), (4,3), (2,1), (3,2), (4,4);
%e A385330    (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);
%e A385330    ...
%e A385330    * The label (2,2) overwrites the (1,1) label (the only known occurrence of an overwrite).
%Y A385330 Cf. A385615.
%K A385330 nonn,tabf
%O A385330 1,3
%A A385330 _Tamas Sandor Nagy_, Jun 25 2025
%E A385330 More terms from _Sean A. Irvine_, Jun 30 2025