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A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.

%I A334704 #40 Jul 09 2025 04:52:30
%S A334704 0,0,0,1,2,8,4,8,20,44,10,20,43,84,152,20,40,78,140,240,372,35,70,130,
%T A334704 224,369,558,824,56,112,200,332,528,780,1132,1544,84,168,293,472,734,
%U A334704 1064,1519,2052,2712,120,240,410,648,988,1408,1982,2652,3480,4448
%N A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.
%C A334704 It follows from the definitions that T(n,k) + A334705(n,k) = A334703(n,k) for 1 <= k <= n.
%H A334704 Tom Duff, <a href="/A334704/a334704.txt">Values of T(n,k) for grids of size up to 50 X 50</a>
%e A334704 Triangle begins:
%e A334704 0,
%e A334704 0, 0,
%e A334704 1, 2, 8,
%e A334704 4, 8, 20, 44,
%e A334704 10, 20, 43, 84, 152,
%e A334704 20, 40, 78, 140, 240, 372,
%e A334704 35, 70, 130, 224, 369, 558, 824,
%e A334704 56, 112, 200, 332, 528, 780, 1132, 1544,
%e A334704 84, 168, 293, 472, 734, 1064, 1519, 2052, 2712,
%e A334704 120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448,
%e A334704 165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992,
%e A334704 ...
%e A334704 This is the lower half of a symmetric array. The full symmetric array begins:
%e A334704 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...
%e A334704 0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, ...
%e A334704 1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, ...
%e A334704 4, 8, 20, 44, 84, 140, 224, 332, 472, 648, 864, 1120, ...
%e A334704 10, 20, 43, 84, 152, 240, 369, 528, 734, 988, 1295, 1652, ...
%e A334704 20, 40, 78, 140, 240, 372, 558, 780, 1064, 1408, 1826, 2304, ...
%e A334704 35, 70, 130, 224, 369, 558, 824, 1132, 1519, 1982, 2542, 3172, ...
%e A334704 56, 112, 200, 332, 528, 780, 1132, 1544, 2052, 2652, 3372, 4172, ...
%e A334704 84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 3480, 4393, 5396, ...
%e A334704 120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448, 5586, 6824, ...
%e A334704 165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992, 8508, ...
%e A334704 220, 440, 732, 1120, 1652, 2304, 3172, 4172, 5396, 6824, 8508, 10332, ...
%e A334704 ...
%Y A334704 This is a companion to the triangles A334703 and A334705.
%Y A334704 Rows (or columns) 1,2,3,4 of the full array are A000292, A007290, A057566, A334706. The main diagonal is A000938.
%K A334704 nonn,tabl
%O A334704 1,5
%A A334704 _N. J. A. Sloane_, Jun 13 2020
%E A334704 Rows 6 onwards from _Tom Duff_. - _N. J. A. Sloane_, Jun 19 2020