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

Author

Hugo Pfoertner and Rainer Rosenthal, May 02 2022

Keywords

• nonn
• tabl

Comments

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

Examples

			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
.
		

Links

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

Crossrefs

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.

Programs

PARI

see Pfoertner link.