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A334705 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three points from an n X k grid of points which are the vertices of a triangle of nonzero area.

%I A334705 #40 Jul 09 2025 04:52:37
%S A334705 0,0,4,0,18,76,0,48,200,516,0,100,412,1056,2148,0,180,738,1884,3820,
%T A334705 6768,0,294,1200,3052,6176,10922,17600,0,448,1824,4628,9352,16516,
%U A334705 26588,40120,0,648,2632,6668,13456,23740,38192,57588,82608,0,900,3650,9232,18612,32812,52758,79508,114000,157252
%N A334705 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three points from an n X k grid of points which are the vertices of a triangle of nonzero area.
%C A334705 It follows from the definitions that T(n,k) + A334704(n,k) = A334703(n,k) for 1 <= k <= n.
%e A334705 Triangle begins:
%e A334705 0,
%e A334705 0, 4,
%e A334705 0, 18, 76,
%e A334705 0, 48, 200, 516,
%e A334705 0, 100, 412, 1056, 2148,
%e A334705 0, 180, 738, 1884, 3820, 6768,
%e A334705 0, 294, 1200, 3052, 6176, 10922, 17600,
%e A334705 0, 448, 1824, 4628, 9352, 16516, 26588, 40120,
%e A334705 0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608,
%e A334705 0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252,
%e A334705 0, 1210, 4900, 12380, 24940, 43934, 70608, 106364, 152456, 210234, 280988,
%e A334705 ...
%e A334705 This is the lower half of a symmetric array. The full symmetric array begins:
%e A334705 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A334705 0, 4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, ...
%e A334705 0, 18, 76, 200, 412, 738, 1200, 1824, 2632, 3650, 4900, 6408, ...
%e A334705 0, 48, 200, 516, 1056, 1884, 3052, 4628, 6668, 9232, 12380, 16176, ...
%e A334705 0, 100, 412, 1056, 2148, 3820, 6176, 9352, 13456, 18612, 24940, 32568, ...
%e A334705 0, 180, 738, 1884, 3820, 6768, 10922, 16516, 23740, 32812, 43934, 57336, ...
%e A334705 0, 294, 1200, 3052, 6176, 10922, 17600, 26588, 38192, 52758, 70608, 92112, ...
%e A334705 0, 448, 1824, 4628, 9352, 16516, 26588, 40120, 57588, 79508, 106364, 138708, ...
%e A334705 0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608, 114000, 152456, 198760, ...
%e A334705 0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252, 210234, 274016 , ...
%e A334705 0, 1210, 4900, 12380, 24940, 43934, 70608, 106364, 152456, 210234, 280988, 366152, ...
%e A334705 ...
%Y A334705 This is a companion to the triangles A334703 and A334704.
%Y A334705 Rows (or columns) 2,3,4,5 of the full array are A045991, A262402, A296367, A334707. The main diagonal is A045996.
%K A334705 nonn,tabl
%O A334705 1,3
%A A334705 _N. J. A. Sloane_, Jun 13 2020
%E A334705 Rows 6 onwards from _Tom Duff_ (see the Duff link in A334704). - _N. J. A. Sloane_, Jun 19 2020