%I #25 Nov 13 2023 07:29:22

%S 1,4,28,136,445,1126,2404,4558,7921,12880,19876,29404,42013,58306,

%T 78940,104626,136129,174268,219916,274000,337501,411454,496948,595126,

%U 707185,834376,978004,1139428,1320061,1521370,1744876,1992154,2264833,2564596,2893180,3252376,3644029,4070038

%N Number of regions formed after n points have been placed in general position on each edge of the triangle in A365929.

%C See A365929 for more information.

%H Scott R. Shannon, <a href="/A367015/a367015.png">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A367015/a367015_1.png">Image for n = 3</a>. Note that although the number of k-gons will vary as the edge points change position the total number of regions will stay constant (at 136 for n = 3) as long as all internal vertices remain simple.

%F Conjecture: a(n) = (9*n^4 - 12*n^3 + 15*n^2 + 4)/4.

%F a(n) = A366932(n) - 3*A366478(n) + 1 by Euler's formula.

%Y Cf. A365929 (internal vertices), A366932 (edges), A366478 (vertices/3).

%K nonn

%O 0,2

%A _Scott R. Shannon_, Nov 01 2023