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Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.
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%I #9 Feb 16 2025 08:34:04

%S 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,

%T 35,56,70,66,60,60,66,70,56,84,112,112,104,100,104,112,112,84,120,168,

%U 176,168,160,160,168,176,168,120,165,240,261,256,245,240,245,256,261,240,165

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

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

%H Andrew Howroyd, <a href="/A360855/b360855.txt">Table of n, a(n) for n = 1..1275</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.

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

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

%e Array begins:

%e =======================================

%e m\n| 1 2 3 4 5 6 7 8 ...

%e ---+-----------------------------------

%e 1 | 0 0 1 4 10 20 35 56 ...

%e 2 | 0 0 2 8 20 40 70 112 ...

%e 3 | 1 2 6 16 35 66 112 176 ...

%e 4 | 4 8 16 32 60 104 168 256 ...

%e 5 | 10 20 35 60 100 160 245 360 ...

%e 6 | 20 40 66 104 160 240 350 496 ...

%e 7 | 35 70 112 168 245 350 490 672 ...

%e 8 | 56 112 176 256 360 496 672 896 ...

%e ...

%o (PARI) T(m, n) = m*binomial(n,3) + n*binomial(m,3)

%Y Main diagonal is A288961.

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

%Y Cf. A286418, A360853.

%K nonn,tabl,easy,changed

%O 1,7

%A _Andrew Howroyd_, Feb 24 2023