A367015 Number of regions formed after n points have been placed in general position on each edge of the triangle in A365929.

1, 4, 28, 136, 445, 1126, 2404, 4558, 7921, 12880, 19876, 29404, 42013, 58306, 78940, 104626, 136129, 174268, 219916, 274000, 337501, 411454, 496948, 595126, 707185, 834376, 978004, 1139428, 1320061, 1521370, 1744876, 1992154, 2264833, 2564596, 2893180, 3252376, 3644029, 4070038

Offset: 0

Author

Scott R. Shannon, Nov 01 2023

Keywords

• nonn

Comments

See A365929 for more information.

Links

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

Crossrefs

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

Formula

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

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