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A353449 T(n,m) is the number of non-congruent quadrilaterals with integer vertex coordinates (x1,1), (n,y2), (x3,m), (1,y4), 1 < x1, x3 < n, 1 < y2, y4 < m, m <= n, such that the 6 distances between the 4 vertices are distinct and (x3-x1)*(y4-y2) > 0, where T(n,m) is a triangle read by rows.

%I A353449 #24 Jun 11 2022 07:53:16
%S A353449 0,0,0,0,0,0,0,1,2,2,0,1,8,15,12,0,3,16,27,49,29,0,7,21,44,71,103,66,
%T A353449 0,9,30,61,106,152,216,131,0,13,41,80,145,213,298,404,245,0,17,55,106,
%U A353449 189,279,383,507,677,373,0,22,69,135,228,345,485,641,848,1054,576
%N A353449 T(n,m) is the number of non-congruent quadrilaterals with integer vertex coordinates (x1,1), (n,y2), (x3,m), (1,y4), 1 < x1, x3 < n, 1 < y2, y4 < m, m <= n, such that the 6 distances between the 4 vertices are distinct and (x3-x1)*(y4-y2) > 0, where T(n,m) is a triangle read by rows.
%C A353449 Property "(x3-x1)*(y4-y2) > 0" holds iff the diagonals (spokes) of the quadrilateral have unequal signs of their slope. In this case the spokes are tilted in the same direction (see example). The framed quadrilateral may be classified as "unisense" iff (x3-x1)*(y4-y2) > 0.
%C A353449 All quadrilaterals of A353532 are classified according to the sign of the product (x3-x1)*(y4-y2) as "all" = "unisense" (> 0) + "contrasense" (< 0) + "static" (= 0). The distinction is invariant under symmetry.
%H A353449 Rainer Rosenthal, <a href="/A353449/b353449.txt">Rows n = 3..100, flattened</a>
%e A353449 The triangle begins
%e A353449     \ m 3   4    5    6    7    8    9   10
%e A353449    n \-------------------------------------
%e A353449    3 |  0   |    |    |    |    |    |    |
%e A353449    4 |  0,  0    |    |    |    |    |    |
%e A353449    5 |  0,  0,   0    |    |    |    |    |
%e A353449    6 |  0,  1,   2,   2    |    |    |    |
%e A353449    7 |  0,  1,   8,  15,  12    |    |    |
%e A353449    8 |  0,  3,  16,  27,  49,  29    |    |
%e A353449    9 |  0,  7,  21,  44,  71, 103,  66    |
%e A353449   10 |  0,  9,  30,  61, 106, 152, 216, 131
%e A353449 .
%e A353449 T(6,4) = 1 because of the third example for (6,4) in A353532:
%e A353449   .
%e A353449    4 | . . . C . .
%e A353449    3 | D . . . . .     A = (x1,1) = (2,1), B = (6,y2) = (6,2)
%e A353449    2 | . . . . . B     C = (x3,4) = (4,4), D = (1,y4) = (1,3)
%e A353449    1 | . A . . . .
%e A353449    y /------------      (x3-x1) * (y4-y2) = (4-2)*(3-2) > 0
%e A353449      x 1 2 3 4 5 6
%e A353449   .
%e A353449 Spokes AC and BD are tilted in the same direction, to the right. The signs of the slopes are unequal: AC has positive slope, and DB has negative slope.
%Y A353449 Cf. A353532 ("all"), A353450 ("contrasense"), A353451 ("static").
%K A353449 nonn,tabl
%O A353449 3,9
%A A353449 _Rainer Rosenthal_, May 13 2022