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

1,5

Comments

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

Links

Table of n, a(n) for n=1..55.

Hugo Pfoertner, PARI program

Example

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.

Prog

(PARI) see link.

Crossrefs

Cf. A353532, A354699, A354701 (diagonal).

Sequence in context: A200684 A345210 A307385 * A046572 A046574 A019743

Adjacent sequences: A354697 A354698 A354699 * A354701 A354702 A354703

Keyword

nonn,tabl

Author

Hugo Pfoertner, Jun 07 2022

Status

approved