A371264 -id:A371264 - cp's OEIS frontend

A371254 Number of vertices formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

1, 2, 4, 4, 15, 7, 70, 64, 208, 220, 550, 397, 1131, 1162, 1981, 2128, 3723, 3259, 5966, 6000, 9010, 9240, 13524, 12745, 19325, 19266, 26434, 26684, 35931, 33301, 47368, 47616, 61216, 61676, 78330, 76789, 98901, 99674, 122656, 123560

Offset: 1

Scott R. Shannon, Mar 16 2024

nonn

Other than for n = 3, 4, and 6, all graphs so far investigated in this sequence contain some internal vertices which are created from the intersections of both 2 and 3 arcs, i.e., no graph contains only simple intersections. This is in contrast to the case where the point pairs are connected by straight lines, see A007569 and A335102, where the odd-n graphs contain only simple intersections. See the attached images.
Other patterns for the intersection arc counts are also seen. If n is divisible by 3 then a central vertex is always present that is created from the crossing of n arcs. If n is divisible by 6, then internal vertices are present that are created from the crossing of 6 arcs. For n = 15 and n = 45, internal vertices are present that are created from the crossing of 5 arcs - it is likely all graphs with n = 15+30*k, k>=0, contain such vertices.
For n = 30, the graph also contains internal vertices that are created from the crossing of 9 arcs. It is likely that all graphs with n divisible by 30 contain such vertices. As the graphs created from the straight line diagonal intersections of the regular n-gon, see A007569, also have the maximum possible line intersection count of 7 when n is divisible by 30, it is plausible that 9 is the maximum possible arc intersection count for any internal vertex, other than the central vertex when n is divisible by 3.
Assuming these patterns hold for all n, is it possible that there is a general formula for the number of vertices, analogous to that in A007569 for the intersections of chords in a regular n-gon?

B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, arXiv:math/9508209v3 [math.MG], 1995-2006.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 15. Note the 5 arc intersections shown in green.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.
Scott R. Shannon, Image for n = 30. Note the 9 arc intersections shown in violet.

Cf. A371253 (regions), A371255 (edges), A371274 (k-gons), A371264 (vertex crossings), A370980 (number of circles), A371373 (complete circles), A007569, A335102, A358746, A331702, A359252.

a(n) = A371255(n) - A371253(n) + 1 by Euler's formula.

A371377 Irregular table read by rows: place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. T(n,k), k>=2, gives the number of vertices formed by the crossing of k arcs.

Original entry on oeis.org

0, 0, 0, 4, 0, 4, 10, 10, 0, 5, 6, 6, 0, 6, 1, 98, 35, 0, 0, 0, 7, 104, 32, 0, 0, 0, 8, 369, 81, 0, 0, 0, 0, 0, 10, 410, 80, 0, 0, 0, 0, 0, 10, 1034, 165, 0, 0, 0, 0, 0, 0, 0, 11, 768, 84, 0, 0, 36, 0, 0, 0, 0, 12, 1, 2288, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 2464, 280, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Scott R. Shannon, Mar 20 2024

See A371373 and A371374 for images of the graphs.

			The table begins:
0;
0;
0, 4;
0, 4;
10, 10, 0, 5;
6, 6, 0, 6, 1;
98, 35, 0, 0, 0, 7;
104, 32, 0, 0, 0, 8;
369, 81, 0, 0, 0, 0, 0, 10;
410, 80, 0, 0, 0, 0, 0, 10;
1034, 165, 0, 0, 0, 0, 0, 0, 0, 11;
768, 84, 0, 0, 36, 0, 0, 0, 0, 12, 1;
2288, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
2464, 280, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14;
4230, 420, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16;
4672, 448, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16;
7990, 680, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17;
7254, 450, 0, 0, 108, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 1;
13148, 969, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19;
13620, 960, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20;
20265, 1323, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22;
21230, 1320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22;
30452, 1771, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23;
29376, 1416, 0, 0, 216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1;
43800, 2300, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25;
45136, 2288, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26;
.
.

Cf. A371373 (vertices), A371374 (regions), A371375 (edges), A371376 (k-gons), A371264, A335102, A007569, A358746, A331702.

Sum of row(n) = A371373(n).

cp's OEIS Frontend

A371254 Number of vertices formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

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