A356984 Number of regions in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

1, 4, 13, 28, 49, 70, 109, 148, 181, 244, 301, 334, 433, 508, 565, 676, 769, 811, 973, 1069, 1165, 1324, 1453, 1534, 1729, 1876, 1957, 2182, 2353, 2446, 2701, 2884, 3013, 3268, 3454, 3538, 3889, 4108, 4261, 4519, 4801, 4960, 5293, 5536, 5668, 6076, 6349, 6502, 6913, 7204, 7405, 7798, 8113

Offset: 0

Author

Scott R. Shannon, Sep 08 2022

Keywords

• nonn

Comments

See A357007 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.
• Talmon Silver, Classification of the intersection points and the number of regions

Crossrefs

Cf. A357007 (vertices), A357008 (edges), A092867, A092098, A332953, A343755.

Formula

a(n) = A357008(n) - A357007(n) + 1 by Euler's formula.

Conjecture: a(n) = 3*n^2 + 1 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.

a(n) = 1 + 3*n + T2(n) + 2*T3(n) + 3*T4(n); a(n) = 1 + 3*n^2 - T3(n) - 3*T4(n), where T2 is the number of internal vertices meeting exactly two segments (these vertices are labeled in the A357007 links as "4 ngons"), T3 is the number of internal vertices meeting exactly three segments ("6 ngons"), and T4 is the number of internal vertices meeting exactly four segments ("8 ngons"). - Talmon Silver, Sep 23 2022