A335682 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of simple interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 6, 6, 0, 0, 10, 12, 12, 10, 0, 0, 15, 18, 24, 18, 15, 0, 0, 21, 27, 36, 36, 27, 21, 0, 0, 28, 36, 54, 54, 54, 36, 28, 0, 0, 36, 48, 72, 82, 82, 72, 48, 36, 0, 0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0, 0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0

Offset: 1

Author

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

Keywords

• nonn
• tabl

Comments

A simple interior vertex is a vertex where exactly two lines cross. In graph theory terms, this is an interior vertex of degree 4.

The case m=n (the main diagonal) is dealt with in A334701. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

This is the only one of the five arrays (A335678-A335682) that does not have an explicit formula.

Let G_m(x) = g.f. for row m. For m <= 9, G_m appears to be a rational function of x with denominator D_m(x), where (writing C_k for the k-th cyclotomic polynomial):

D_3 = D_4 = C_1^3*C_2

D_5 = C_1^3*C_2*C_4

D_6 = C_1^3*C_2*C_4*C_5

D_7 = C_1^3*C_2*C_3*C_4*C_5*C_6

D_8 = D_9 = C_1^3*C_2*C_3*C_4*C_5*C_6*C_7

Examples

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
0, 3, 6, 12, 18, 27, 36, 48, 60, 75, 90, 108, ...
0, 6, 12, 24, 36, 54, 72, 96, 120, 150, 180, 216, ...
0, 10, 18, 36, 54, 82, 108, 144, 180, 226, 270, 324, ...
0, 15, 27, 54, 82, 124, 163, 217, 272, 342, 408, 489, ...
0, 21, 36, 72, 108, 163, 214, 286, 358, 451, 536, 642, ...
0, 28, 48, 96, 144, 217, 286, 382, 478, 602, 715, 856, ...
0, 36, 60, 120, 180, 272, 358, 478, 598, 754, 894, 1070, ...
0, 45, 75, 150, 226, 342, 451, 602, 754, 950, 1126, 1347, ...
0, 55, 90, 180, 270, 408, 536, 715, 894, 1126, 1334, 1597, ...
0, 66, 108, 216, 324, 489, 642, 856, 1070, 1347, 1597, 1912, ...
...
The initial antidiagonals are:
0
0, 0
0, 1, 0
0, 3, 3, 0
0, 6, 6, 6, 0
0, 10, 12, 12, 10, 0
0, 15, 18, 24, 18, 15, 0
0, 21, 27, 36, 36, 27, 21, 0
0, 28, 36, 54, 54, 54, 36, 28, 0
0, 36, 48, 72, 82, 82, 72, 48, 36, 0
0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0
0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0
...
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 antidiagonals)
• Index entries for sequences related to stained glass windows

Crossrefs

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.

For the diagonal case see A306302, A331755, A334701.