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

0,1

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.

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

Adjacent sequences: A357004 A357005 A357006 * A357008 A357009 A357010

Keyword

nonn

Author

Scott R. Shannon, Sep 08 2022

Status

approved