login
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.
6

%I #23 Sep 10 2022 21:05:02

%S 3,6,15,30,51,66,111,150,171,246,303,312,435,510,543,678,771,765,975,

%T 1059,1131,1326,1455,1488,1731,1878,1899,2178,2355,2376,2703,2886,

%U 2955,3270,3444,3420,3891,4110,4191,4485,4803,4878,5295,5526,5544,6078,6351,6396,6915,7206,7311,7794,8115

%N 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.

%C See A356984 for further images.

%H Scott R. Shannon, <a href="/A357007/b357007.txt">Table of n, a(n) for n = 0..250</a>

%H Scott R. Shannon, <a href="/A357007/a357007.png">Image for n = 1</a>.

%H Scott R. Shannon, <a href="/A357007/a357007_1.png">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A357007/a357007_2.png">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A357007/a357007_3.png">Image for n = 5</a>. This is the first term that forms intersections with non-simple vertices.

%H Scott R. Shannon, <a href="/A357007/a357007_4.png">Image for n = 10</a>.

%H Scott R. Shannon, <a href="/A357007/a357007_5.png">Image for n = 50</a>.

%H Scott R. Shannon, <a href="/A357007/a357007_6.png">Image for n = 100</a>.

%H Scott R. Shannon, <a href="/A357007/a357007_7.png">Image for n = 200</a>.

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

%F 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.

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

%K nonn

%O 0,1

%A _Scott R. Shannon_, Sep 08 2022