A367121 Place n points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of regions in the resulting planar graph.

4, 67, 406, 1441, 3796, 8299, 15982, 28081, 46036, 71491, 106294, 152497, 212356, 288331, 383086, 499489, 640612, 809731, 1010326, 1246081, 1520884, 1838827, 2204206, 2621521, 3095476, 3630979, 4233142, 4907281, 5658916, 6493771, 7417774, 8437057, 9557956, 10787011, 12130966

Offset: 0

Author

Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

Keywords

• nonn

Comments

"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.

Note that although the number of k-gons in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices remain simple.

Links

• Scott R. Shannon, Image for n = 0.
• Scott R. Shannon, Image for n = 1.
• Scott R. Shannon, Image for n = 2.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.

Crossrefs

Cf. A334698 (vertices), A367122 (edges), A255011, A367118.

Formula

Conjecture: a(n) = (17/2)*n^4 + 19*n^3 + (43/2)*n^2 + 14*n + 4.

a(n) = A367122(n) - A334698(n+1) + 1 by Euler's formula.