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A353451 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 A353451 #27 Apr 21 2023 06:29:25
%S A353451 0,0,0,0,2,0,1,6,6,4,1,8,10,19,12,4,15,20,39,48,40,4,16,25,41,52,89,
%T A353451 47,10,30,39,66,86,135,144,105,10,31,41,75,91,140,142,212,106,18,49,
%U A353451 67,107,134,203,220,308,319,214,18,49,67,109,144,210,227,325,334,458,228
%N A353451 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 A353451 Property "(x3-x1)*(y4-y2) = 0" holds iff one of the diagonals (spokes) of the quadrilateral is parallel to the x-axis or to the y-axis, i.e. not tilted (see example). The framed quadrilateral may be classified as "static" iff (x3-x1)*(y4-y2) = 0.
%C A353451 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 A353451 Rainer Rosenthal, <a href="/A353451/b353451.txt">Rows n = 3..100, flattened</a>
%e A353451 The triangle begins
%e A353451 .
%e A353451     \ m 3   4    5    6    7    8    9   10
%e A353451    n \-------------------------------------
%e A353451    3 |  0   |    |    |    |    |    |    |
%e A353451    4 |  0,  0    |    |    |    |    |    |
%e A353451    5 |  0,  2,   0    |    |    |    |    |
%e A353451    6 |  1,  6,   6,   4    |    |    |    |
%e A353451    7 |  1,  8,  10,  19,  12    |    |    |
%e A353451    8 |  4, 15,  20,  39,  48,  40    |    |
%e A353451    9 |  4, 16,  25,  41,  52,  89,  47    |
%e A353451   10 | 10, 30,  39,  66,  86, 135, 144, 105
%e A353451 .
%e A353451 T(5,4) = a(5) = 2: See first 2 examples for (5,4) in A353532.
%e A353451   .
%e A353451      4 | . C . . .
%e A353451      3 | . . . . .     A = (x1,1) = (2,1), B = (5,y2) = (5,2)
%e A353451      2 | D . . . B     C = (x3,4) = (2,4), D = (1,y4) = (1,2)
%e A353451      1 | . A . . .
%e A353451      y /----------      (x3-x1) * (y4-y2) = (2-2)*(2-2) = 0
%e A353451        x 1 2 3 4 5
%e A353451   .
%e A353451      4 | . C . . .
%e A353451      3 | . . . . B     A = (x1,1) = (2,1), B = (5,y2) = (5,3)
%e A353451      2 | D . . . .     C = (x3,4) = (2,4), D = (1,y4) = (1,2)
%e A353451      1 | . A . . .
%e A353451      y /----------      (x3-x1) * (y4-y2) = (2-2)*(2-3) = 0
%e A353451        x 1 2 3 4 5
%e A353451   .
%e A353451 T(5,4) = 2 since these are the only static configurations of A353532(5,4). Spoke AC is not tilted, but parallel to the y-axis. First example: spoke DB is not tilted, but parallel to the x-axis. Second example: spoke DB is not parallel to the x-axis, but tilted to the left. We have (x3-x1)*(y4-y2) = 0 in both cases, so these framed quadrilaterals have the "static" property.
%Y A353451 Cf. A353532 ("all"), A353449 ("unisense"), A353450 ("contrasense").
%K A353451 nonn,tabl
%O A353451 3,5
%A A353451 _Rainer Rosenthal_, May 13 2022