%I #22 Aug 04 2020 08:54:52
%S 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,
%T 21,27,36,36,27,21,0,0,28,36,54,54,54,36,28,0,0,36,48,72,82,82,72,48,
%U 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
%N 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.
%C A simple interior vertex is a vertex where exactly two lines cross. In graph theory terms, this is an interior vertex of degree 4.
%C The case m=n (the main diagonal) is dealt with in A334701. A306302 has illustrations for the diagonal case for m = 1 to 15.
%C Also A335678 has colored illustrations for many values of m and n.
%C This is the only one of the five arrays (A335678-A335682) that does not have an explicit formula.
%C 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):
%C D_3 = D_4 = C_1^3*C_2
%C D_5 = C_1^3*C_2*C_4
%C D_6 = C_1^3*C_2*C_4*C_5
%C D_7 = C_1^3*C_2*C_3*C_4*C_5*C_6
%C D_8 = D_9 = C_1^3*C_2*C_3*C_4*C_5*C_6*C_7
%H Lars Blomberg, <a href="/A335682/b335682.txt">Table of n, a(n) for n = 1..9870</a> (the first 140 antidiagonals)
%H <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>
%e The initial rows of the array are:
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
%e 0, 3, 6, 12, 18, 27, 36, 48, 60, 75, 90, 108, ...
%e 0, 6, 12, 24, 36, 54, 72, 96, 120, 150, 180, 216, ...
%e 0, 10, 18, 36, 54, 82, 108, 144, 180, 226, 270, 324, ...
%e 0, 15, 27, 54, 82, 124, 163, 217, 272, 342, 408, 489, ...
%e 0, 21, 36, 72, 108, 163, 214, 286, 358, 451, 536, 642, ...
%e 0, 28, 48, 96, 144, 217, 286, 382, 478, 602, 715, 856, ...
%e 0, 36, 60, 120, 180, 272, 358, 478, 598, 754, 894, 1070, ...
%e 0, 45, 75, 150, 226, 342, 451, 602, 754, 950, 1126, 1347, ...
%e 0, 55, 90, 180, 270, 408, 536, 715, 894, 1126, 1334, 1597, ...
%e 0, 66, 108, 216, 324, 489, 642, 856, 1070, 1347, 1597, 1912, ...
%e ...
%e The initial antidiagonals are:
%e 0
%e 0, 0
%e 0, 1, 0
%e 0, 3, 3, 0
%e 0, 6, 6, 6, 0
%e 0, 10, 12, 12, 10, 0
%e 0, 15, 18, 24, 18, 15, 0
%e 0, 21, 27, 36, 36, 27, 21, 0
%e 0, 28, 36, 54, 54, 54, 36, 28, 0
%e 0, 36, 48, 72, 82, 82, 72, 48, 36, 0
%e 0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0
%e 0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0
%e ...
%Y This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
%Y For the diagonal case see A306302, A331755, A334701.
%K nonn,tabl
%O 1,8
%A _Lars Blomberg_, _Scott R. Shannon_, and _N. J. A. Sloane_, Jun 28 2020