A357007 Number of vertices in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

3, 6, 15, 30, 51, 66, 111, 150, 171, 246, 303, 312, 435, 510, 543, 678, 771, 765, 975, 1059, 1131, 1326, 1455, 1488, 1731, 1878, 1899, 2178, 2355, 2376, 2703, 2886, 2955, 3270, 3444, 3420, 3891, 4110, 4191, 4485, 4803, 4878, 5295, 5526, 5544, 6078, 6351, 6396, 6915, 7206, 7311, 7794, 8115

Offset: 0

Author

Scott R. Shannon, Sep 08 2022

Keywords

• nonn

Comments

See A356984 for further images.

Links

• Scott R. Shannon, Table of n, a(n) for n = 0..250
• 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 = 5. This is the first term that forms intersections with non-simple vertices.
• Scott R. Shannon, Image for n = 10.
• Scott R. Shannon, Image for n = 50.
• Scott R. Shannon, Image for n = 100.
• Scott R. Shannon, Image for n = 200.

Crossrefs

Cf. A356984 (regions), A357008 (edges), A092866, A091908, A333026, A344657.

Formula

a(n) = A357008(n) - A356984(n) + 1 by Euler's formula.

Conjecture: a(n) = 3*n^2 + 3 for equilateral triangles that only contain simple vertices when cut by n internal equilateral triangles. This is never the case if (n + 1) mod 3 = 0 for n > 3.