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.

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, 0, 9, 30, 61, 106, 152, 216, 131, 0, 13, 41, 80, 145, 213, 298, 404, 245, 0, 17, 55, 106, 189, 279, 383, 507, 677, 373, 0, 22, 69, 135, 228, 345, 485, 641, 848, 1054, 576

Offset: 3

Author

Rainer Rosenthal, May 13 2022

Keywords

• nonn
• tabl

Comments

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.

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.

Examples

			The triangle begins
    \ m 3   4    5    6    7    8    9   10
   n \-------------------------------------
   3 |  0   |    |    |    |    |    |    |
   4 |  0,  0    |    |    |    |    |    |
   5 |  0,  0,   0    |    |    |    |    |
   6 |  0,  1,   2,   2    |    |    |    |
   7 |  0,  1,   8,  15,  12    |    |    |
   8 |  0,  3,  16,  27,  49,  29    |    |
   9 |  0,  7,  21,  44,  71, 103,  66    |
  10 |  0,  9,  30,  61, 106, 152, 216, 131
.
T(6,4) = 1 because of the third example for (6,4) in A353532:
  .
   4 | . . . C . .
   3 | D . . . . .     A = (x1,1) = (2,1), B = (6,y2) = (6,2)
   2 | . . . . . B     C = (x3,4) = (4,4), D = (1,y4) = (1,3)
   1 | . A . . . .
   y /------------      (x3-x1) * (y4-y2) = (4-2)*(3-2) > 0
     x 1 2 3 4 5 6
  .
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.
		

Links

• Rainer Rosenthal, Rows n = 3..100, flattened

Crossrefs

Cf. A353532 ("all"), A353450 ("contrasense"), A353451 ("static").