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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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

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

EXAMPLE

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

...

CROSSREFS

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

For the diagonal case see A306302, A331755, A334701.

Sequence in context: A245320 A330341 A152893 * A335681 A297978 A298629

Adjacent sequences: A335679 A335680 A335681 * A335683 A335684 A335685

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified January 30 12:30 EST 2023. Contains 359945 sequences. (Running on oeis4.)