A353533 -id:A353533 - cp's OEIS frontend

A353532 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.

0, 0, 0, 0, 3, 1, 1, 7, 12, 11, 1, 11, 26, 52, 40, 4, 23, 50, 94, 147, 105, 4, 30, 69, 127, 198, 301, 190, 10, 49, 103, 192, 302, 444, 583, 379, 10, 58, 127, 244, 387, 576, 754, 1039, 616, 18, 84, 180, 329, 509, 756, 989, 1334, 1680, 987, 18, 94, 209, 389, 611, 910, 1203, 1618, 2052, 2581, 1426

Offset: 3

Hugo Pfoertner and Rainer Rosenthal, May 02 2022

T(n,m) is a triangle, read by rows.

			The triangle begins
    \ m 3   4    5    6    7    8    9   10
   n \-------------------------------------
   3 |  0,  |    |    |    |    |    |    |
   4 |  0,  0,   |    |    |    |    |    |
   5 |  0,  3,   1,   |    |    |    |    |
   6 |  1,  7,  12,  11,   |    |    |    |
   7 |  1, 11,  26,  52,  40,   |    |    |
   8 |  4, 23,  50,  94, 147, 105,   |    |
   9 |  4, 30,  69, 127, 198, 301, 190,   |
  10 | 10, 49, 103, 192, 302, 444, 583, 379
.
.
   4 | . C . . .    There are six squared distances.
   3 | . . . . .    They are arranged as follows:
   2 | D . . . B      AB-BC-CD-DA  (counterclockwise)
   1 | . A . . .      AC X DB      (across)
   y /----------    Here: AB = 3^2 + 1^2 = 10,
     x 1 2 3 4 5          BC = 13, CD = 5, DA = 2,
.                         AC =  9, DB = 16
      10-13-5-2  <==== yielding this
      9 X 16     <==== description
.
.
T(5,4) = a(5) = 3:
.
   4 | . X . . .     4 | . X . . .     4 | . . X . .
   3 | . . . . .     3 | . . . . X     3 | . . . . X
   2 | X . . . X     2 | X . . . .     2 | X . . . .
   1 | . X . . .     1 | . X . . .     1 | . X . . .
   y /----------     y /----------     y /----------
     x 1 2 3 4 5       x 1 2 3 4 5       x 1 2 3 4 5
.
      10-13-5-2          13-10-5-2          13-5-8-2
      9 X 16             9 X 17             10 X 17
.
T(5,5) = a(6) = A353447(5) = 1:
.
   5 | . . . X .
   4 | . . . . .
   3 | . . . . X    13-5-18-2
   2 | X . . . .    20 X 17
   1 | . X . . .
   y /----------
     x 1 2 3 4 5
.
T(6,3) = a(7) = 1:
.
   3 | . . . X . .
   2 | X . . . . X    17-5-10-2
   1 | . X . . . .    8 X 25
   y /------------
     x 1 2 3 4 5 6
.
T(6,4) = a(8) = 7:
.
   4 | . X . . . .   4 | . X . . . .   4 | . . X . . .   4 | . . . X . .
   3 | . . . . . .   3 | . . . . . X   3 | . . . . . .   3 | X . . . . .
   2 | X . . . . X   2 | X . . . . .   2 | X . . . . X   2 | . . . . . X
   1 | . X . . . .   1 | . X . . . .   1 | . X . . . .   1 | . X . . . .
   y /------------   y /------------   y /------------   y /------------
     x 1 2 3 4 5 6     x 1 2 3 4 5 6     x 1 2 3 4 5 6     x 1 2 3 4 5 6
.
       17-20-5-2         20-17-5-2         17-13-8-2         17-8-10-5
       9 X 25            9 X 26            10 X 25           13 X 26
.
   4 | . . . . X .   4 | . . X . . .   4 | . . X . . .
   3 | . . . . . .   3 | . . . . . .   3 | . . . . . X
   2 | X . . . . X   2 | X . . . . X   2 | X . . . . .
   1 | . X . . . .   1 | . . X . . .   1 | . . X . . .
   y /------------   y /------------   y /------------
     x 1 2 3 4 5 6     x 1 2 3 4 5 6     x 1 2 3 4 5 6
.
       17-5-20-2         10-13-8-5         13-10-8-5
       18 X 25           9 X 25            9 X 26
.

Rainer Rosenthal, Rows n = 3..100, flattened
Hugo Pfoertner, PARI program

Cf. A353447 (diagonal), A353449, A353450, A353451, A353533, A354700.

The general case without excluding the corners of the grid rectangle is covered in A354700 and A354701.

PARI
```
see Pfoertner link.
```

A354488 T(w,h) with 3 <= h < w is the number of quadrilaterals as defined in A353532 with diagonals intersecting at the same angle theta as the diagonals of the grid rectangle with side lengths w > h, where T(w,h) is a triangle read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0

Offset: 4

Hugo Pfoertner and Rainer Rosenthal, May 28 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.
Quadrilaterals with this property cannot occur for rectangles with h = 2 and for rectangles with h = w. Thus the triangle is given without the column h = 2 and the diagonal h = w.
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 intersection angle of the rectangle's diagonals is delta = 2*phi, where phi is the angle between a diagonal and a longer side of the grid rectangle. So tan(delta) = 2*tan(phi)/(1 - tan(phi)^2) where tan(phi) = h/w, i.e., tan(delta) = 2*w*h/(w^2 - h^2).

			The triangle begins:
   4: 0,
   5: 0,0,
   6: 0,0, 0,
   7: 0,0, 0, 0,
   8: 0,3, 0, 0, 0,
   9: 4,0, 0, 0, 0, 0,
  10: 0,0, 0, 0, 0, 0, 0,
  11: 0,0, 0, 0, 0, 0, 0, 0,
  12: 0,0, 0, 3, 0,11, 0, 0,0,
  13: 0,0, 0, 0, 0, 0, 0, 0,0,  0,
  14: 0,0, 0, 0,12, 0, 0, 0,0,  0,0,
  15: 0,0, 0, 0, 0, 0, 0,32,0,  0,0, 0,
  16: 0,0, 0, 0, 0,23, 0, 0,0,  0,0, 0,  0,
  17: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,
  18: 0,0, 0,33, 0, 0,51, 0,0, 53,0, 0,  0, 0,0,
  19: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,
  20: 0,0, 0, 0, 0, 0, 0, 0,0, 46,0, 0,  0, 0,0, 0,0,
  21: 0,0, 0, 0,18, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,
  22: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,
  23: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,0,
  24: 0,0, 0, 0, 0,53, 0, 0,0,107,0, 0,  0,57,0,91,0,0,  0,0,0,
  25: 0,0,24, 0, 0, 0, 0, 0,0,  0,0, 0,108, 0,0, 0,0,0,  0,0,0,0,
  26: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,0,0,0,0,
  27: 0,0, 0, 0, 0, 0,55, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,0,0,0,0,0,
  28: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0,57,  0, 0,0, 0,0,0,182,0,0,0,0,0,0,
  29: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,0,0,0,0,0,0,0
   n  ------------------------------------------------------------------
   m: 3 4  5  6  7  8  9 10 .  12 . 14  15 16 . 18 . .  21 . . . . . . 28
.
T(8,4) = 3, tan(theta) = 4/3 = tan(2*phi).
Intersection angle of diagonals of the grid rectangle:
tan(2*phi) = 2 *(1/2) / (1 - (1/2)^2) = 1 / (3/4) = 4/3, with tan(phi) = 4/8 = 1/2.
.
  4 | . . . . . C . . .   4 | . . . . . C . . .   4 | . . . . . . C . .
  3 | . . . . . . . . .   3 | . . . . . . . . .   3 | . . . . . . . . .
  2 | . . . . . . . . .   3 | D . . . . . . . B   2 | . . . . . . . . .
  1 | D . . . . . . . B   1 | . . . . . . . . .   1 | D . . . . . . . B
  0 | . . A . . . . . .   0 | . . A . . . . . .   0 | . . . A . . . . .
  y /------------------   y /------------------   y /------------------
    x 0 1 2 3 4 5 6 7 8     x 0 1 2 3 4 5 6 7 8     x 0 1 2 3 4 5 6 7 8
.
T(9,3) = 4, tan(theta) = 3/4 = tan(2*phi).
tan(phi) = 3/9 = 1/3, tan(2*phi) = 2*(1/3)/(1 - (1/3)^2) = (2/3)/(8/9) = 18/24 = 3/4.
.
  3 | . . . . . C . . . .       3 | . . . . . C . . . .
  2 | . . . . . . . . . .       2 | D . . . . . . . . B
  1 | D . . . . . . . . B       1 | . . . . . . . . . .
  0 | . A . . . . . . . .       0 | . A . . . . . . . .
  y /--------------------       y /--------------------
    x 0 1 2 3 4 5 6 7 8 9         x 0 1 2 3 4 5 6 7 8 9
.
  3 | . . . . . . C . . .       3 | . . . . . . C . . .
  2 | . . . . . . . . . .       2 | D . . . . . . . . B
  1 | D . . . . . . . . B       1 | . . . . . . . . . .
  0 | . . A . . . . . . .       0 | . . A . . . . . . .
  y /--------------------       y /--------------------
    x 0 1 2 3 4 5 6 7 8 9         x 0 1 2 3 4 5 6 7 8 9
.
T(12,6) = 3, with slopes of diagonals of quadrilateral against y = 0: sAC, sDB, sAC = 6/2 = 3, sDB = 4/12 = 1/3, angle difference theta = sAC - sDB.
Using tan(alpha - beta) = (tan(alpha) - tan(beta))/(1 + tan(alpha)*tan(beta)), tan(theta) = (sAC - sBD) / (1 + sAC*sBD) = (3 - 1/3)/( 1 + 1 ) = 4/3.
tan(phi) = 6/12 = 1/2; tan(2*phi) = 2*(1/2)/(1 - (1/2)^2) = 1/(3/4) = 4/3.
.
  6 | . . . C . . . . . . . . .       6 | . . . . C . . . . . . . .
  5 | . . . . . . . . . . . . B       5 | . . . . . . . . . . . . B
  4 | . . . . . . . . . . . . .       4 | . . . . . . . . . . . . .
  3 | . . . . . . . . . . . . .       3 | . . . . . . . . . . . . .
  2 | . . . . . . . . . . . . .       2 | . . . . . . . . . . . . .
  1 | D . . . . . . . . . . . .       1 | D . . . . . . . . . . . .
  0 | . A . . . . . . . . . . .       0 | . . A . . . . . . . . . .
  y /--------------------------       y /--------------------------
    x 0 1 2 3 4 5 6 7 8 9 0 1 2         x 0 1 2 3 4 5 6 7 8 9 0 1 2
.
  6 | . . . . . . C . . . . . .
  5 | . . . . . . . . . . . . B
  4 | . . . . . . . . . . . . .
  3 | . . . . . . . . . . . . .
  2 | . . . . . . . . . . . . .
  1 | D . . . . . . . . . . . .
  0 | . . . . A . . . . . . . .
  y /--------------------------
    x 0 1 2 3 4 5 6 7 8 9 0 1 2

Hugo Pfoertner, PARI program to print list of nonzero sequence terms.

Cf. A353532, A353533.

A354489 provides the widths of those grid rectangles for which no inserted quadrilaterals with matching angles of the diagonals exist, i.e., all terms = 0 in a row of the triangle.

\\ See link. The program a354488(w1,w2) prints a list of the nonzero terms [w, d, T_a353532(w+1,d+1), T(w,d)] in the range w1 <= w <= w2.

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A353532 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A354488 T(w,h) with 3 <= h < w is the number of quadrilaterals as defined in A353532 with diagonals intersecting at the same angle theta as the diagonals of the grid rectangle with side lengths w > h, where T(w,h) is a triangle read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI