cp's OEIS Frontend

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.

%I A367121 #18 Nov 13 2023 18:37:10
%S A367121 4,67,406,1441,3796,8299,15982,28081,46036,71491,106294,152497,212356,
%T A367121 288331,383086,499489,640612,809731,1010326,1246081,1520884,1838827,
%U A367121 2204206,2621521,3095476,3630979,4233142,4907281,5658916,6493771,7417774,8437057,9557956,10787011,12130966
%N 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.
%C A367121 "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.
%C A367121 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.
%H A367121 Scott R. Shannon, <a href="/A367121/a367121.png">Image for n = 0</a>.
%H A367121 Scott R. Shannon, <a href="/A367121/a367121_1.png">Image for n = 1</a>.
%H A367121 Scott R. Shannon, <a href="/A367121/a367121_2.png">Image for n = 2</a>.
%H A367121 Scott R. Shannon, <a href="/A367121/a367121_3.png">Image for n = 3</a>.
%H A367121 Scott R. Shannon, <a href="/A367121/a367121_4.png">Image for n = 4</a>.
%F A367121 Conjecture: a(n) = (17/2)*n^4 + 19*n^3 + (43/2)*n^2 + 14*n + 4.
%F A367121 a(n) = A367122(n) - A334698(n+1) + 1 by Euler's formula.
%Y A367121 Cf. A334698 (vertices), A367122 (edges), A255011, A367118.
%K A367121 nonn
%O A367121 0,1
%A A367121 _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 05 2023