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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.
5
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.
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
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner, Jun 07 2022
STATUS
approved