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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.
6
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
0,1
COMMENTS
See A356984 for further images.
LINKS
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.
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.
CROSSREFS
Cf. A356984 (regions), A357008 (edges), A092866, A091908, A333026, A344657.
Sequence in context: A077449 A356666 A152232 * A183038 A141023 A242172
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Sep 08 2022
STATUS
approved