A354490 T(w,h) with 2 <= h <= w is the number of quadrilaterals as defined in A353532 with diagonals intersecting at integer coordinates, where T(w,h) is a triangle read by rows.

0, 0, 0, 0, 1, 0, 1, 3, 1, 0, 0, 3, 3, 4, 4, 3, 6, 6, 6, 12, 0, 2, 6, 7, 9, 15, 13, 6, 6, 10, 12, 12, 30, 18, 27, 8, 4, 11, 11, 12, 24, 25, 33, 41, 18, 10, 17, 21, 17, 36, 24, 35, 32, 38, 0, 8, 17, 19, 21, 51, 43, 65, 84, 87, 57, 62, 15, 24, 31, 25, 49, 31, 48, 45, 53, 33, 76, 0

Offset: 2

Author

Hugo Pfoertner, May 30 2022

Keywords

• nonn
• tabl

Comments

The integer coordinates of the 4 vertices of the quadrilateral are (x1,0), (w,y2), (x3,h), (0,y4), 0 < x1, x3 < w, 0 < y2, y4 < h, such that the 6 distances between the 4 vertices are distinct.

The relationship to A353532 is that the number of lattice points n X m is used there, while here the side lengths of the lattice rectangle w = n - 1 and h = m - 1 are used.

Examples

			The triangle begins, with corresponding terms of A353532 shown in parenthesis:
   \ d 2       3       4        5        6        7        8       9
  w \---------------------------------------------------------------------
  2 |  0 ( 0)  |       |        |        |        |        |       |
  3 |  0 ( 0)  0 ( 0)  |        |        |        |        |       |
  4 |  0 ( 0)  1 ( 3)  0 (  1)  |        |        |        |       |
  5 |  1 ( 1)  3 ( 7)  1 ( 12)  0 ( 11)  |        |        |       |
  6 |  0 ( 1)  3 (11)  3 ( 26)  4 ( 52)  4 ( 40)  |        |       |
  7 |  3 ( 4)  6 (23)  6 ( 50)  6 ( 94) 12 (147)  0 (105)  |       |
  8 |  2 ( 4)  6 (30)  7 ( 69)  9 (127) 15 (198) 13 (301)  6 (190) |
  9 |  6 (10) 10 (49) 12 (103) 12 (192) 30 (302) 18 (444) 27 (583) 8 (379)
.
Only 1 = T(4,3) of the 3 = T_a353532(5,4) quadrilaterals has diagonals AC, BD whose intersection point S has integer coordinates:
.
   3 | . C . . .     3 | . C . . .     3 | . . C . .
   2 | . . . . .     2 | . . . . B     2 | . . . . B
   1 | D S . . B     1 | D . . . .     1 | D . . . .
   0 | . A . . .     0 | . A . . .     0 | . A . . .
   y /----------     y /----------     y /----------
     x 0 1 2 3 4       x 0 1 2 3 4       x 0 1 2 3 4
        S=(1,1)          S=(1,5/4)     S=(16/11,15/11)
.
T(5,2) = T_a353532(6,3) = 1:
.
   2 | . . . C . .
   1 | D . S . . B
   0 | . A . . . .
   y /------------
     x 0 1 2 3 4 5
        S=(2,1)
.
T(5,3) = 3 of the T_a353532(6,4) = 7 intersection points S of the diagonals AC, BD have integer coordinates:
.
  3 | . C . . . .   3 | . C . . . .   3 | . . C . . .   3 | . . . C . .
  2 | . . . . . .   2 | . . . . . B   2 | . . . . . .   2 | D . . . . .
  1 | D S . . . B   1 | D . . . . .   1 | D . . . . B   1 | . . . . . B
  0 | . A . . . .   0 | . A . . . .   0 | . A . . . .   0 | . A . . . .
  y /------------   y /------------   y /------------   y /------------
    x 0 1 2 3 4 5     x 0 1 2 3 4 5     x 0 1 2 3 4 5     x 0 1 2 3 4 5
       S=(1,1)           S=(1,6/5)         S=(4/3,1)     S=(35/17,27/17)
.
  3 | . . . . C .   3 | . . C . . .   3 | . . C . . .
  2 | . . . . . .   2 | . . . . . .   2 | . . . . . B
  1 | D . S . . B   1 | D . S . . B   1 | D . . . . .
  0 | . A . . . .   0 | . . A . . .   0 | . . A . . .
  y /------------   y /------------   y /------------
    x 0 1 2 3 4 5     x 0 1 2 3 4 5     x 0 1 2 3 4 5
       S=(2,1)           S=(2,1)           S=(2,7/5)
		

Links

• Hugo Pfoertner, PARI program to print sequence terms.

Crossrefs

Cf. A353532, A353533, A354488.

A354491 is the diagonal of the triangle.

Programs

PARI

\\ See link. The program a354490 (w1, w2) prints the terms for the rows w1 .. w2. An auxiliary function sinter is defined to determine the rational intersection point of the diagonals.