This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354700 #12 Dec 19 2024 11:53:22
%S A354700 0,0,0,1,4,5,2,16,36,21,8,33,69,116,71,13,52,126,201,317,181,22,84,
%T A354700 191,299,445,639,366,28,110,249,373,581,839,1105,585,43,157,330,529,
%U A354700 806,1094,1463,1856,1009,50,190,407,653,1014,1360,1853,2295,2958,1562
%N 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.
%C A354700 T(w,h) is a triangle read by rows, 1 <= h <= w.
%H A354700 Hugo Pfoertner, <a href="/A354700/a354700.gp.txt">PARI program</a>
%e A354700 The triangle begins:
%e A354700 0;
%e A354700 0, 0;
%e A354700 1, 4, 5;
%e A354700 2, 16, 36, 21;
%e A354700 8, 33, 69, 116, 71;
%e A354700 13, 52, 126, 201, 317, 181;
%e A354700 22, 84, 191, 299, 445, 639, 366;
%e A354700 28, 110, 249, 373, 581, 839, 1105, 585
%e A354700 .
%e A354700 T(3,1) = 1:
%e A354700 1 | D . . C Squared distances:
%e A354700 0 | A . B . Sides: AB = 4, BC = 2, CD = 9, DA = 1;
%e A354700 y /-------- Diagonals: AC = 10, BD = 5.
%e A354700 x 0 1 2 3
%e A354700 .
%e A354700 T(3,2) = 4:
%e A354700 2 | D . . C Squared distances:
%e A354700 1 | . . . . Sides: AB = 1, BC = 8, CD = 9, DA = 4;
%e A354700 0 | A B . . Diagonals: AC = 13, BD = 5.
%e A354700 y /--------
%e A354700 x 0 1 2 3
%e A354700 2 | . . . D Squared distances:
%e A354700 1 | . . . C Sides: AB = 4, BC = 2, CD = 1, DA = 13;
%e A354700 0 | A . B . Diagonals: AC = 10, BD = 5.
%e A354700 y /--------
%e A354700 x 0 1 2 3
%e A354700 2 | . . D . Squared distances:
%e A354700 1 | . . . C Sides: AB = 9, BC = 1, CD = 2, DA = 8;
%e A354700 0 | A . . B Diagonals: AC = 10, BD = 5.
%e A354700 y /--------
%e A354700 x 0 1 2 3
%e A354700 2 | . . C . Squared distances:
%e A354700 1 | D . . B Sides: AB = 10, BC = 2, CD = 5, DA = 1;
%e A354700 0 | A . . . Diagonals: AC = 8, BD = 9.
%e A354700 y /--------
%e A354700 x 0 1 2 3
%e A354700 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.
%e A354700 .
%e A354700 T(3,3) = 5:
%e A354700 3 | . D . C 3 | . . . C 3 | . . . D 3 | . D . . 3 | . . D .
%e A354700 2 | . . . . 2 | D . . . 2 | . . . . 2 | . . . C 2 | . . . .
%e A354700 1 | . . . . 1 | . . . . 1 | . . . C 1 | . . . . 1 | . . . C
%e A354700 0 | A B . . 0 | A B . . 0 | A B . . 0 | A B . . 0 | A . B .
%e A354700 y /-------- y /-------- y /-------- y /-------- y /--------
%e A354700 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
%e A354700 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.
%o A354700 (PARI) \\ See link.
%Y A354700 Cf. A353532, A354699, A354701 (diagonal).
%K A354700 nonn,tabl
%O A354700 1,5
%A A354700 _Hugo Pfoertner_, Jun 07 2022