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 A354490 #8 Dec 19 2024 11:53:22 %S A354490 0,0,0,0,1,0,1,3,1,0,0,3,3,4,4,3,6,6,6,12,0,2,6,7,9,15,13,6,6,10,12, %T A354490 12,30,18,27,8,4,11,11,12,24,25,33,41,18,10,17,21,17,36,24,35,32,38,0, %U A354490 8,17,19,21,51,43,65,84,87,57,62,15,24,31,25,49,31,48,45,53,33,76,0 %N A354490 T(w,h) with 2 <= h <= w is the number of quadrilaterals as defined in A353532 with diagonals intersecting at integer coordinates, where T(w,h) is a triangle read by rows. %C A354490 The integer coordinates of the 4 vertices of the quadrilateral are (x1,0), (w,y2), (x3,h), (0,y4), 0 < x1, x3 < w, 0 < y2, y4 < h, such that the 6 distances between the 4 vertices are distinct. %C A354490 The relationship to A353532 is that the number of lattice points n X m is used there, while here the side lengths of the lattice rectangle w = n - 1 and h = m - 1 are used. %H A354490 Hugo Pfoertner, <a href="/A354490/a354490_1.gp.txt">PARI program to print sequence terms</a>. %e A354490 The triangle begins, with corresponding terms of A353532 shown in parenthesis: %e A354490 \ d 2 3 4 5 6 7 8 9 %e A354490 w \--------------------------------------------------------------------- %e A354490 2 | 0 ( 0) | | | | | | | %e A354490 3 | 0 ( 0) 0 ( 0) | | | | | | %e A354490 4 | 0 ( 0) 1 ( 3) 0 ( 1) | | | | | %e A354490 5 | 1 ( 1) 3 ( 7) 1 ( 12) 0 ( 11) | | | | %e A354490 6 | 0 ( 1) 3 (11) 3 ( 26) 4 ( 52) 4 ( 40) | | | %e A354490 7 | 3 ( 4) 6 (23) 6 ( 50) 6 ( 94) 12 (147) 0 (105) | | %e A354490 8 | 2 ( 4) 6 (30) 7 ( 69) 9 (127) 15 (198) 13 (301) 6 (190) | %e A354490 9 | 6 (10) 10 (49) 12 (103) 12 (192) 30 (302) 18 (444) 27 (583) 8 (379) %e A354490 . %e A354490 Only 1 = T(4,3) of the 3 = T_a353532(5,4) quadrilaterals has diagonals AC, BD whose intersection point S has integer coordinates: %e A354490 . %e A354490 3 | . C . . . 3 | . C . . . 3 | . . C . . %e A354490 2 | . . . . . 2 | . . . . B 2 | . . . . B %e A354490 1 | D S . . B 1 | D . . . . 1 | D . . . . %e A354490 0 | . A . . . 0 | . A . . . 0 | . A . . . %e A354490 y /---------- y /---------- y /---------- %e A354490 x 0 1 2 3 4 x 0 1 2 3 4 x 0 1 2 3 4 %e A354490 S=(1,1) S=(1,5/4) S=(16/11,15/11) %e A354490 . %e A354490 T(5,2) = T_a353532(6,3) = 1: %e A354490 . %e A354490 2 | . . . C . . %e A354490 1 | D . S . . B %e A354490 0 | . A . . . . %e A354490 y /------------ %e A354490 x 0 1 2 3 4 5 %e A354490 S=(2,1) %e A354490 . %e A354490 T(5,3) = 3 of the T_a353532(6,4) = 7 intersection points S of the diagonals AC, BD have integer coordinates: %e A354490 . %e A354490 3 | . C . . . . 3 | . C . . . . 3 | . . C . . . 3 | . . . C . . %e A354490 2 | . . . . . . 2 | . . . . . B 2 | . . . . . . 2 | D . . . . . %e A354490 1 | D S . . . B 1 | D . . . . . 1 | D . . . . B 1 | . . . . . B %e A354490 0 | . A . . . . 0 | . A . . . . 0 | . A . . . . 0 | . A . . . . %e A354490 y /------------ y /------------ y /------------ y /------------ %e A354490 x 0 1 2 3 4 5 x 0 1 2 3 4 5 x 0 1 2 3 4 5 x 0 1 2 3 4 5 %e A354490 S=(1,1) S=(1,6/5) S=(4/3,1) S=(35/17,27/17) %e A354490 . %e A354490 3 | . . . . C . 3 | . . C . . . 3 | . . C . . . %e A354490 2 | . . . . . . 2 | . . . . . . 2 | . . . . . B %e A354490 1 | D . S . . B 1 | D . S . . B 1 | D . . . . . %e A354490 0 | . A . . . . 0 | . . A . . . 0 | . . A . . . %e A354490 y /------------ y /------------ y /------------ %e A354490 x 0 1 2 3 4 5 x 0 1 2 3 4 5 x 0 1 2 3 4 5 %e A354490 S=(2,1) S=(2,1) S=(2,7/5) %o A354490 (PARI) \\ See link. The program a354490 (w1, w2) prints the terms for the rows w1 .. w2. An auxiliary function sinter is defined to determine the rational intersection point of the diagonals. %Y A354490 Cf. A353532, A353533, A354488. %Y A354490 A354491 is the diagonal of the triangle. %K A354490 nonn,tabl %O A354490 2,8 %A A354490 _Hugo Pfoertner_, May 30 2022