A354488 -id:A354488 - cp's OEIS frontend

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

Hugo Pfoertner, May 30 2022

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.

			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)

Hugo Pfoertner, PARI program to print sequence terms.

Cf. A353532, A353533, A354488.

A354491 is the diagonal of the triangle.

\\ 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.

A354489 Widths w of w X h grid rectangles with w > h such that no quadrilaterals with 2 < h < w as defined in A353532 exist, whose angle between their diagonals is equal to the angle between the diagonals of the grid rectangle.

Original entry on oeis.org

4, 5, 6, 7, 10, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 89, 92, 94, 95, 97

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, May 28 2022

The relationship to A353532 is given by the fact 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.

			See A354488.

Cf. A353532, A354488 for more information.

There are many common terms with A325160.

cp's OEIS Frontend

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.

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