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.

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, 0, 10, 34, 65, 110, 157, 223, 143, 0, 14, 45, 89, 151, 223, 314, 423, 265, 0, 18, 58, 116, 186, 274, 386, 519, 684, 400, 0, 23, 73, 145, 239, 355, 491, 652, 870, 1069, 622

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 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.

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,  1,   1    |    |    |    |    |
   6 |  0,  0,   4,   5    |    |    |    |
   7 |  0,  2,   8,  18,  16    |    |    |
   8 |  0,  5,  14,  28,  50,  36    |    |
   9 |  0,  7,  23,  42,  75, 109,  77    |
  10 |  0, 10,  34,  65, 110, 157, 223, 143
.
T(5,4) = 1 because of the third example for (5,4) in A353532.
  .
   4 | . . C . .
   3 | . . . . B     A = (x1,1) = (2,1), B = (5,y2) = (5,3)
   2 | D . . . .     C = (x3,4) = (3,4), D = (1,y4) = (1,2)
   1 | . A . . .
   y /----------      (x3-x1) * (y4-y2) = (3-2)*(2-3) < 0
     x 1 2 3 4 5
  .
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).
  .
T(5,5) = 1 because of the third example for (5,5) in A353532.
  .
   5 | . . . C .
   4 | . . . . .     A = (x1,1) = (2,1), B = (5,y2) = (5,3)
   3 | . . . . B     C = (x3,5) = (4,5), D = (1,y4) = (1,2)
   2 | D . . . .
   1 | . A . . .     (x3-x1) * (y4-y2) = (4-2)*(2-3) < 0
   y /----------
     x 1 2 3 4 5
  .
Spokes AC and DB are tilted in different directions ("contrasense") like in the example before.
		

Links

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

Crossrefs

Cf. A353532 ("all"), A353449 ("unisense"), A353451 ("static").