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

0,2

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

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

Crossrefs

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

Sequence in context: A272455 A220745 A298017 * A307272 A056107 A155433

Adjacent sequences: A356981 A356982 A356983 * A356985 A356986 A356987

Keyword

nonn

Author

Scott R. Shannon, Sep 08 2022

Status

approved