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

3,9

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.

Links

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

Example

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.

Crossrefs

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

Sequence in context: A199390 A270914 A048837 * A087492 A366362 A199967

Adjacent sequences: A353447 A353448 A353449 * A353451 A353452 A353453

Keyword

nonn,tabl

Author

Rainer Rosenthal, May 13 2022

Status

approved