A353532 -id:A353532 - cp's OEIS frontend

A353533 T(n,m) with 4 <= m < n is the number of quadrilaterals in A353532 with perpendicular diagonals, where T(n,m) is a triangle read by rows.

1, 2, 1, 2, 2, 3, 3, 3, 4, 6, 3, 5, 5, 8, 9, 4, 4, 6, 12, 12, 12, 4, 4, 12, 8, 11, 15, 14, 5, 5, 8, 10, 15, 15, 20, 18, 5, 5, 8, 27, 15, 33, 32, 26, 25, 6, 6, 10, 11, 17, 17, 23, 22, 29, 29, 6, 6, 10, 12, 48, 18, 24, 29, 30, 42, 34, 7, 7, 16, 14, 21, 21, 41, 69, 34

Offset: 5

Hugo Pfoertner and Rainer Rosenthal, May 04 2022

			The quadrilaterals counted in A353532 with m = 3 or m = n cannot have perpendicular diagonals, and are therefore omitted in the triangle of this sequence.
.
    \ m 3   4   5   6   7   8   9  10  11
   n \-----------------------------------
   3 |  0,  |   |   |   |   |   |   |   |
   4 |  0,  0,  |   |   |   |   |   |   |
   5 |  0,  1,  0,  |   |   |   |   |   |
   6 |  0,  2,  1,  0,  |   |   |   |   |
   7 |  0,  2,  2,  3,  0,  |   |   |   |
   8 |  0,  3,  3,  4,  6,  0,  |   |   |
   9 |  0,  3,  5,  5,  8,  9,  0,  |   |
  10 |  0,  4,  4,  6, 12, 12, 12,  0,  |
  11 |  0,  4,  4, 12,  8, 11, 15, 14,  0
.
T(5,4) = a(1) = 1:
.
   4 | . C . . .      Squared distances denoted
   3 | . . . . .      as in examples A353532:
   2 | D . . . B
   1 | . A . . .       AB-BC-CD-DA (around)
   y /----------       AC X DB     (across)
     x 1 2 3 4 5
.
      10-13-5-2
      9 X 16
.
T(6,4) = a(2) = 2:
.
   4 | . X . . . .     4 | . . X . . .
   3 | . . . . . .     3 | . . . . . .
   2 | X . . . . X     2 | X . . . . X
   1 | . X . . . .     1 | . . X . . .
   y /------------     y /------------
     x 1 2 3 4 5 6       x 1 2 3 4 5 6
.
      17-20-5-2           10-13-8-5
      9 X 25              9 X 25
.
T(6,5) = a(3) = 1:
.
   5 | . . X . . .
   4 | . . . . . .
   3 | . . . . . .     10-18-13-5
   2 | X . . . . X     16 X 25
   1 | . . X . . .
   y /------------
     x 1 2 3 4 5 6
.
T(9,5) = a(12) = 5;
3 quadrilaterals with diagonals parallel to the grid axes:
.
   5 | . X . . . . . . .   5 | . . X . . . . . .   5 | . . . X . . . . .
   4 | . . . . . . . . .   4 | . . . . . . . . .   4 | . . . . . . . . .
   3 | . . . . . . . . .   3 | . . . . . . . . .   3 | . . . . . . . . .
   2 | X . . . . . . . X   2 | X . . . . . . . X   2 | X . . . . . . . X
   1 | . X . . . . . . .   1 | . . X . . . . . .   1 | . . . X . . . . .
   y /------------------   y /------------------   y /------------------
     x 1 2 3 4 5 6 7 8 9     x 1 2 3 4 5 6 7 8 9     x 1 2 3 4 5 6 7 8 9
.
         50-58-10-2              37-45-13-5              26-34-18-10
         16 X 64                 16 X 64                 16 X 64
.
The 2 quadrilaterals with diagonals not aligned with the grid axes are the smallest example of this type:
.
.
   5 | . X . . . . . . .   5 | . . X . . . . . .
   4 | . . . . . . . . X   4 | . . . . . . . . X
   3 | . . . . . . . . .   3 | . . . . . . . . .
   2 | X . . . . . . . .   2 | X . . . . . . . .
   1 | . . X . . . . . .   1 | . . . X . . . . .
   y /------------------   y /------------------
     x 1 2 3 4 5 6 7 8 9     x 1 2 3 4 5 6 7 8 9
.
         45-50-10-5              34-37-13-10
         17 X 68                 17 X 68
.

Cf. A353447, A353532.

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.

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.

A353447 a(n) is the number of tetrapods standing on the four edges of an n X n grid, so that no two feet are the same distance apart and no foot is on a corner. Tetrapods with congruent footprints are counted only once.

Original entry on oeis.org

0, 0, 1, 11, 40, 105, 190, 379, 616, 987, 1426, 2139, 2964, 4130, 5403, 7180, 9155, 11716, 14458, 18092, 22037, 26808, 31793, 38343, 45060, 53184, 61613, 71878, 82466, 95368, 108195, 123790, 140040, 158457, 177405, 200020, 223039, 248769, 275214, 306411, 337645

Offset: 3

Rainer Rosenthal, Apr 20 2022

nonn

If we name the tetrapod's footprints "mini-frame", we can say that mini-frames span their grid, i.e., there is no smaller grid for them. Every corner-less set of points with distinct distances in a smallest possible n X n grid contains at least one mini-frame.

			  .
     . C .           a(3) = 0              . . . C .
     D . B   <===  since AB = CD           . . . . .
     . A .         is forbidden            . . . . B
                                           . . . . .
                        . C . .            D . . . .
      a(4) = 0  ===>    ? . . .            . A . . .
    (there is no        ? . . B         ______________
     space for D)       . A . .            a(5) = 1
                                     (No other solutions)
  .
    . . . . .           The tetrapod has 6 distinct
    D . . . .           squared distances 4, 5, 10,
    . . . . C   <=====  13, 17, 18, but it uses only
    . . . . .           three edges of the 5 X 5 grid.
    . A . B .           (Not allowed.)
  .

Hugo Pfoertner, Table of n, a(n) for n = 3..200

Cf. A193838, A271490, A335232, A351699, A351700, A353532.

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

a(23) and beyond from Hugo Pfoertner, Apr 20 2022

A353449 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.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 1, 8, 15, 12, 0, 3, 16, 27, 49, 29, 0, 7, 21, 44, 71, 103, 66, 0, 9, 30, 61, 106, 152, 216, 131, 0, 13, 41, 80, 145, 213, 298, 404, 245, 0, 17, 55, 106, 189, 279, 383, 507, 677, 373, 0, 22, 69, 135, 228, 345, 485, 641, 848, 1054, 576

Offset: 3

Rainer Rosenthal, May 13 2022

Property "(x3-x1)*(y4-y2) > 0" holds iff the diagonals (spokes) of the quadrilateral have unequal signs of their slope. In this case the spokes are tilted in the same direction (see example). The framed quadrilateral may be classified as "unisense" 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.

			The triangle begins
    \ m 3   4    5    6    7    8    9   10
   n \-------------------------------------
   3 |  0   |    |    |    |    |    |    |
   4 |  0,  0    |    |    |    |    |    |
   5 |  0,  0,   0    |    |    |    |    |
   6 |  0,  1,   2,   2    |    |    |    |
   7 |  0,  1,   8,  15,  12    |    |    |
   8 |  0,  3,  16,  27,  49,  29    |    |
   9 |  0,  7,  21,  44,  71, 103,  66    |
  10 |  0,  9,  30,  61, 106, 152, 216, 131
.
T(6,4) = 1 because of the third example for (6,4) in A353532:
  .
   4 | . . . C . .
   3 | D . . . . .     A = (x1,1) = (2,1), B = (6,y2) = (6,2)
   2 | . . . . . B     C = (x3,4) = (4,4), D = (1,y4) = (1,3)
   1 | . A . . . .
   y /------------      (x3-x1) * (y4-y2) = (4-2)*(3-2) > 0
     x 1 2 3 4 5 6
  .
Spokes AC and BD are tilted in the same direction, to the right. The signs of the slopes are unequal: AC has positive slope, and DB has negative slope.

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

Cf. A353532 ("all"), A353450 ("contrasense"), A353451 ("static").

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.

Original entry on oeis.org

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

Rainer Rosenthal, May 13 2022

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.

			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.

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

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

A354700 T(w,h) is the number of non-congruent quadrilaterals whose vertices with integer coordinates (x_i, y_i) all lie on the perimeter of a rectangle of width w and height h, with no 3 points on the same edge of the rectangle, max(x_i) - min(x_i) = w and max(y_i) - min(y_i) = h, such that the 6 distances between the 4 vertices are distinct.

Original entry on oeis.org

0, 0, 0, 1, 4, 5, 2, 16, 36, 21, 8, 33, 69, 116, 71, 13, 52, 126, 201, 317, 181, 22, 84, 191, 299, 445, 639, 366, 28, 110, 249, 373, 581, 839, 1105, 585, 43, 157, 330, 529, 806, 1094, 1463, 1856, 1009, 50, 190, 407, 653, 1014, 1360, 1853, 2295, 2958, 1562

Offset: 1

Hugo Pfoertner, Jun 07 2022

T(w,h) is a triangle read by rows, 1 <= h <= w.

			The triangle begins:
   0;
   0,   0;
   1,   4,   5;
   2,  16,  36,  21;
   8,  33,  69, 116,  71;
  13,  52, 126, 201, 317, 181;
  22,  84, 191, 299, 445, 639,  366;
  28, 110, 249, 373, 581, 839, 1105, 585
.
T(3,1) = 1:
  1 | D . . C  Squared distances:
  0 | A . B .  Sides: AB = 4, BC = 2, CD = 9, DA = 1;
  y /--------  Diagonals: AC = 10, BD = 5.
    x 0 1 2 3
.
T(3,2) = 4:
  2 | D . . C  Squared distances:
  1 | . . . .  Sides: AB = 1, BC = 8, CD = 9, DA = 4;
  0 | A B . .  Diagonals: AC = 13, BD = 5.
  y /--------
    x 0 1 2 3
  2 | . . . D  Squared distances:
  1 | . . . C  Sides: AB = 4, BC = 2, CD = 1, DA = 13;
  0 | A . B .  Diagonals: AC = 10, BD = 5.
  y /--------
    x 0 1 2 3
  2 | . . D .  Squared distances:
  1 | . . . C  Sides: AB = 9, BC = 1, CD = 2, DA = 8;
  0 | A . . B  Diagonals: AC = 10, BD = 5.
  y /--------
    x 0 1 2 3
  2 | . . C .  Squared distances:
  1 | D . . B  Sides: AB = 10, BC = 2, CD = 5, DA = 1;
  0 | A . . .  Diagonals: AC = 8, BD = 9.
  y /--------
    x 0 1 2 3
The last 2 quadrilaterals have the same set {1, 2, 5, 8, 9, 10} of squared distances, but with different allocation of sides and diagonals.
.
T(3,3) = 5:
  3 | . D . C    3 | . . . C    3 | . . . D    3 | . D . .    3 | . . D .
  2 | . . . .    2 | D . . .    2 | . . . .    2 | . . . C    2 | . . . .
  1 | . . . .    1 | . . . .    1 | . . . C    1 | . . . .    1 | . . . C
  0 | A B . .    0 | A B . .    0 | A B . .    0 | A B . .    0 | A . B .
  y /--------    y /--------    y /--------    y /--------    y /--------
    x 0 1 2 3      x 0 1 2 3      x 0 1 2 3      x 0 1 2 3      x 0 1 2 3
Quadrilaterals Q2 and Q3 have the same set {1, 4, 5, 10, 13, 18} of squared distances, but the allocation of sides and diagonals differ: Squared diagonals are AC, BD {18,5} in Q2, and {10,13} in Q3.

Hugo Pfoertner, PARI program

Cf. A353532, A354699, A354701 (diagonal).

PARI
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A353451 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.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 6, 6, 4, 1, 8, 10, 19, 12, 4, 15, 20, 39, 48, 40, 4, 16, 25, 41, 52, 89, 47, 10, 30, 39, 66, 86, 135, 144, 105, 10, 31, 41, 75, 91, 140, 142, 212, 106, 18, 49, 67, 107, 134, 203, 220, 308, 319, 214, 18, 49, 67, 109, 144, 210, 227, 325, 334, 458, 228

Offset: 3

Rainer Rosenthal, May 13 2022

Property "(x3-x1)*(y4-y2) = 0" holds iff one of the diagonals (spokes) of the quadrilateral is parallel to the x-axis or to the y-axis, i.e. not tilted (see example). The framed quadrilateral may be classified as "static" 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.

			The triangle begins
.
    \ m 3   4    5    6    7    8    9   10
   n \-------------------------------------
   3 |  0   |    |    |    |    |    |    |
   4 |  0,  0    |    |    |    |    |    |
   5 |  0,  2,   0    |    |    |    |    |
   6 |  1,  6,   6,   4    |    |    |    |
   7 |  1,  8,  10,  19,  12    |    |    |
   8 |  4, 15,  20,  39,  48,  40    |    |
   9 |  4, 16,  25,  41,  52,  89,  47    |
  10 | 10, 30,  39,  66,  86, 135, 144, 105
.
T(5,4) = a(5) = 2: See first 2 examples for (5,4) in A353532.
  .
     4 | . C . . .
     3 | . . . . .     A = (x1,1) = (2,1), B = (5,y2) = (5,2)
     2 | D . . . B     C = (x3,4) = (2,4), D = (1,y4) = (1,2)
     1 | . A . . .
     y /----------      (x3-x1) * (y4-y2) = (2-2)*(2-2) = 0
       x 1 2 3 4 5
  .
     4 | . C . . .
     3 | . . . . B     A = (x1,1) = (2,1), B = (5,y2) = (5,3)
     2 | D . . . .     C = (x3,4) = (2,4), D = (1,y4) = (1,2)
     1 | . A . . .
     y /----------      (x3-x1) * (y4-y2) = (2-2)*(2-3) = 0
       x 1 2 3 4 5
  .
T(5,4) = 2 since these are the only static configurations of A353532(5,4). Spoke AC is not tilted, but parallel to the y-axis. First example: spoke DB is not tilted, but parallel to the x-axis. Second example: spoke DB is not parallel to the x-axis, but tilted to the left. We have (x3-x1)*(y4-y2) = 0 in both cases, so these framed quadrilaterals have the "static" property.

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

Cf. A353532 ("all"), A353449 ("unisense"), A353450 ("contrasense").

A354701 Diagonal of the triangle A354700.

Original entry on oeis.org

0, 0, 5, 21, 71, 181, 366, 585, 1009, 1562, 2312, 3206, 4490, 5967, 7939, 10023, 12913, 15900, 19951, 24153, 29483, 35227, 42039, 49103, 57998, 67518, 78426, 90010, 103631, 117759, 134551, 150970, 171440, 192305, 215740, 239549, 268137, 296993, 329001, 361740, 400113

Offset: 1

Hugo Pfoertner, Jun 07 2022

nonn

			See A354700.

Hugo Pfoertner, Table of n, a(n) for n = 1..76

Cf. A353447, A353532.

a(n) = A354700(n, n).

cp's OEIS Frontend

A353533 T(n,m) with 4 <= m < n is the number of quadrilaterals in A353532 with perpendicular diagonals, where T(n,m) is a triangle read by rows.

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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.

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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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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.

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A353447 a(n) is the number of tetrapods standing on the four edges of an n X n grid, so that no two feet are the same distance apart and no foot is on a corner. Tetrapods with congruent footprints are counted only once.

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A353449 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.

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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.

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A353451 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.

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A354701 Diagonal of the triangle A354700.

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