login
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.
5
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
0,1
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.
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.
CROSSREFS
Cf. A334698 (vertices), A367122 (edges), A255011, A367118.
Sequence in context: A048828 A225940 A003360 * A225772 A329478 A241001
KEYWORD
nonn
AUTHOR
STATUS
approved