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A353450 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 A353450 #24 Jun 11 2022 07:53:37
%S A353450 0,0,0,0,1,1,0,0,4,5,0,2,8,18,16,0,5,14,28,50,36,0,7,23,42,75,109,77,
%T A353450 0,10,34,65,110,157,223,143,0,14,45,89,151,223,314,423,265,0,18,58,
%U A353450 116,186,274,386,519,684,400,0,23,73,145,239,355,491,652,870,1069,622
%N A353450 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 A353450 Property "(x3-x1)*(y4-y2) < 0" holds iff the diagonals (spokes) of the quadrilateral are concurrent, i.e., their slopes are both positive or both negative. In this case the spokes are tilted in a different sense: clockwise and counterclockwise (see example). The framed quadrilateral may be classified as "contrasense" iff (x3-x1)*(y4-y2) < 0.
%C A353450 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 A353450 Rainer Rosenthal, <a href="/A353450/b353450.txt">Rows n = 3..100, flattened</a>
%e A353450 The triangle begins
%e A353450     \ m 3   4    5    6    7    8    9   10
%e A353450    n \-------------------------------------
%e A353450    3 |  0   |    |    |    |    |    |    |
%e A353450    4 |  0,  0    |    |    |    |    |    |
%e A353450    5 |  0,  1,   1    |    |    |    |    |
%e A353450    6 |  0,  0,   4,   5    |    |    |    |
%e A353450    7 |  0,  2,   8,  18,  16    |    |    |
%e A353450    8 |  0,  5,  14,  28,  50,  36    |    |
%e A353450    9 |  0,  7,  23,  42,  75, 109,  77    |
%e A353450   10 |  0, 10,  34,  65, 110, 157, 223, 143
%e A353450 .
%e A353450 T(5,4) = 1 because of the third example for (5,4) in A353532.
%e A353450   .
%e A353450    4 | . . C . .
%e A353450    3 | . . . . B     A = (x1,1) = (2,1), B = (5,y2) = (5,3)
%e A353450    2 | D . . . .     C = (x3,4) = (3,4), D = (1,y4) = (1,2)
%e A353450    1 | . A . . .
%e A353450    y /----------      (x3-x1) * (y4-y2) = (3-2)*(2-3) < 0
%e A353450      x 1 2 3 4 5
%e A353450   .
%e A353450 Spokes AC and BD are tilted in different directions ("contrasense"). AC has positive slope and is tilted to the right (clockwise), DB also has same sign of slope, but is tilted to the left (counterclockwise).
%e A353450   .
%e A353450 T(5,5) = 1 because of the third example for (5,5) in A353532.
%e A353450   .
%e A353450    5 | . . . C .
%e A353450    4 | . . . . .     A = (x1,1) = (2,1), B = (5,y2) = (5,3)
%e A353450    3 | . . . . B     C = (x3,5) = (4,5), D = (1,y4) = (1,2)
%e A353450    2 | D . . . .
%e A353450    1 | . A . . .     (x3-x1) * (y4-y2) = (4-2)*(2-3) < 0
%e A353450    y /----------
%e A353450      x 1 2 3 4 5
%e A353450   .
%e A353450 Spokes AC and DB are tilted in different directions ("contrasense") like in the example before.
%Y A353450 Cf. A353532 ("all"), A353449 ("unisense"), A353451 ("static").
%K A353450 nonn,tabl
%O A353450 3,9
%A A353450 _Rainer Rosenthal_, May 13 2022