A360855 Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.

0, 0, 0, 1, 0, 1, 4, 2, 2, 4, 10, 8, 6, 8, 10, 20, 20, 16, 16, 20, 20, 35, 40, 35, 32, 35, 40, 35, 56, 70, 66, 60, 60, 66, 70, 56, 84, 112, 112, 104, 100, 104, 112, 112, 84, 120, 168, 176, 168, 160, 160, 168, 176, 168, 120, 165, 240, 261, 256, 245, 240, 245, 256, 261, 240, 165

Offset: 1

Author

Andrew Howroyd, Feb 24 2023

Keywords

• nonn
• tabl
• easy

Comments

A triangle is a clique of size 3. Also, a 3-cycle.

Examples

			Array begins:
=======================================
m\n|  1   2   3   4   5   6   7   8 ...
---+-----------------------------------
1  |  0   0   1   4  10  20  35  56 ...
2  |  0   0   2   8  20  40  70 112 ...
3  |  1   2   6  16  35  66 112 176 ...
4  |  4   8  16  32  60 104 168 256 ...
5  | 10  20  35  60 100 160 245 360 ...
6  | 20  40  66 104 160 240 350 496 ...
7  | 35  70 112 168 245 350 490 672 ...
8  | 56 112 176 256 360 496 672 896 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• Eric Weisstein's World of Mathematics, Rook Graph.

Crossrefs

Main diagonal is A288961.

Rows n=1..3 are A000292(n-2), A007290, A060354.

Cf. A286418, A360853.

Programs

PARI

T(m, n) = m*binomial(n,3) + n*binomial(m,3)

Formula

T(m,n) = m*binomial(n,3) + n*binomial(m,3).

T(m,n) = T(n,m).