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.

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

Author

Hugo Pfoertner, Jun 07 2022

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
		

Links

• Hugo Pfoertner, PARI program

Crossrefs

Cf. A353532, A354699, A354701 (diagonal).

Programs

PARI

\\ See link.