A371373 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. The sequence gives the total number of vertices formed.

1, 2, 4, 4, 25, 19, 140, 144, 460, 500, 1210, 901, 2587, 2758, 4696, 5136, 8687, 7831, 14136, 14600, 21610, 22572, 32246, 31033, 46125, 47450, 63748, 65772, 86565, 82051, 114824, 117760, 148930, 152796, 190820, 189973, 241055, 247038, 300028, 306840, 369943, 367711, 451586, 459448

Offset: 1

Author

Scott R. Shannon, Mar 20 2024

Keywords

• nonn

Comments

The number of circles that cross to form the intersections follows a similar pattern to that seen in A371254; see that sequence for further information. The details of the crossing counts are given in A371377.

Links

• Scott R. Shannon, Image for n = 3. In this and other images the initial circle on which the n points are placed is drawn in a thicker white line for clarity.
• 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. Note the 6 circle intersections shown in blue.
• Scott R. Shannon, Image for n = 15. Note the 5 circle 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 circle intersections shown in violet.

Crossrefs

Cf. A371374 (regions), A371375 (edges), A371376 (k-gons), A371377 (vertex crossings), A371254, A007569, A358746, A331702.

Formula

a(n) = A371375(n) - A371374(n) + 1 by Euler's formula.