A353447 -id:A353447 - 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.
```

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).

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.

Original entry on oeis.org

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.

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.

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A354701 Diagonal of the triangle A354700.

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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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A354491 Diagonal of the triangle A354490.

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