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

%I A360855 #9 Feb 16 2025 08:34:04
%S A360855 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 A360855 35,56,70,66,60,60,66,70,56,84,112,112,104,100,104,112,112,84,120,168,
%U A360855 176,168,160,160,168,176,168,120,165,240,261,256,245,240,245,256,261,240,165
%N A360855 Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.
%C A360855 A triangle is a clique of size 3. Also, a 3-cycle.
%H A360855 Andrew Howroyd, <a href="/A360855/b360855.txt">Table of n, a(n) for n = 1..1275</a>
%H A360855 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%F A360855 T(m,n) = m*binomial(n,3) + n*binomial(m,3).
%F A360855 T(m,n) = T(n,m).
%e A360855 Array begins:
%e A360855 =======================================
%e A360855 m\n|  1   2   3   4   5   6   7   8 ...
%e A360855 ---+-----------------------------------
%e A360855 1  |  0   0   1   4  10  20  35  56 ...
%e A360855 2  |  0   0   2   8  20  40  70 112 ...
%e A360855 3  |  1   2   6  16  35  66 112 176 ...
%e A360855 4  |  4   8  16  32  60 104 168 256 ...
%e A360855 5  | 10  20  35  60 100 160 245 360 ...
%e A360855 6  | 20  40  66 104 160 240 350 496 ...
%e A360855 7  | 35  70 112 168 245 350 490 672 ...
%e A360855 8  | 56 112 176 256 360 496 672 896 ...
%e A360855   ...
%o A360855 (PARI) T(m, n) = m*binomial(n,3) + n*binomial(m,3)
%Y A360855 Main diagonal is A288961.
%Y A360855 Rows n=1..3 are A000292(n-2), A007290, A060354.
%Y A360855 Cf. A286418, A360853.
%K A360855 nonn,tabl,easy
%O A360855 1,7
%A A360855 _Andrew Howroyd_, Feb 24 2023