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%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
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