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 A354488 #10 Dec 19 2024 11:53:22 %S A354488 0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, %T A354488 0,0,0,0,0,3,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0, %U A354488 0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0 %N A354488 T(w,h) with 3 <= h < w is the number of quadrilaterals as defined in A353532 with diagonals intersecting at the same angle theta as the diagonals of the grid rectangle with side lengths w > h, where T(w,h) is a triangle read by rows. %C A354488 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 A354488 Quadrilaterals with this property cannot occur for rectangles with h = 2 and for rectangles with h = w. Thus the triangle is given without the column h = 2 and the diagonal h = w. %C A354488 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. %C A354488 The intersection angle of the rectangle's diagonals is delta = 2*phi, where phi is the angle between a diagonal and a longer side of the grid rectangle. So tan(delta) = 2*tan(phi)/(1 - tan(phi)^2) where tan(phi) = h/w, i.e., tan(delta) = 2*w*h/(w^2 - h^2). %H A354488 Hugo Pfoertner, <a href="/A354488/a354488.gp.txt">PARI program to print list of nonzero sequence terms</a>. %e A354488 The triangle begins: %e A354488 4: 0, %e A354488 5: 0,0, %e A354488 6: 0,0, 0, %e A354488 7: 0,0, 0, 0, %e A354488 8: 0,3, 0, 0, 0, %e A354488 9: 4,0, 0, 0, 0, 0, %e A354488 10: 0,0, 0, 0, 0, 0, 0, %e A354488 11: 0,0, 0, 0, 0, 0, 0, 0, %e A354488 12: 0,0, 0, 3, 0,11, 0, 0,0, %e A354488 13: 0,0, 0, 0, 0, 0, 0, 0,0, 0, %e A354488 14: 0,0, 0, 0,12, 0, 0, 0,0, 0,0, %e A354488 15: 0,0, 0, 0, 0, 0, 0,32,0, 0,0, 0, %e A354488 16: 0,0, 0, 0, 0,23, 0, 0,0, 0,0, 0, 0, %e A354488 17: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0, %e A354488 18: 0,0, 0,33, 0, 0,51, 0,0, 53,0, 0, 0, 0,0, %e A354488 19: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0, %e A354488 20: 0,0, 0, 0, 0, 0, 0, 0,0, 46,0, 0, 0, 0,0, 0,0, %e A354488 21: 0,0, 0, 0,18, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0, %e A354488 22: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0, 0, %e A354488 23: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0, 0,0, %e A354488 24: 0,0, 0, 0, 0,53, 0, 0,0,107,0, 0, 0,57,0,91,0,0, 0,0,0, %e A354488 25: 0,0,24, 0, 0, 0, 0, 0,0, 0,0, 0,108, 0,0, 0,0,0, 0,0,0,0, %e A354488 26: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0, 0,0,0,0,0, %e A354488 27: 0,0, 0, 0, 0, 0,55, 0,0, 0,0, 0, 0, 0,0, 0,0,0, 0,0,0,0,0,0, %e A354488 28: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0,57, 0, 0,0, 0,0,0,182,0,0,0,0,0,0, %e A354488 29: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0, 0,0,0,0,0,0,0,0 %e A354488 n ------------------------------------------------------------------ %e A354488 m: 3 4 5 6 7 8 9 10 . 12 . 14 15 16 . 18 . . 21 . . . . . . 28 %e A354488 . %e A354488 T(8,4) = 3, tan(theta) = 4/3 = tan(2*phi). %e A354488 Intersection angle of diagonals of the grid rectangle: %e A354488 tan(2*phi) = 2 *(1/2) / (1 - (1/2)^2) = 1 / (3/4) = 4/3, with tan(phi) = 4/8 = 1/2. %e A354488 . %e A354488 4 | . . . . . C . . . 4 | . . . . . C . . . 4 | . . . . . . C . . %e A354488 3 | . . . . . . . . . 3 | . . . . . . . . . 3 | . . . . . . . . . %e A354488 2 | . . . . . . . . . 3 | D . . . . . . . B 2 | . . . . . . . . . %e A354488 1 | D . . . . . . . B 1 | . . . . . . . . . 1 | D . . . . . . . B %e A354488 0 | . . A . . . . . . 0 | . . A . . . . . . 0 | . . . A . . . . . %e A354488 y /------------------ y /------------------ y /------------------ %e A354488 x 0 1 2 3 4 5 6 7 8 x 0 1 2 3 4 5 6 7 8 x 0 1 2 3 4 5 6 7 8 %e A354488 . %e A354488 T(9,3) = 4, tan(theta) = 3/4 = tan(2*phi). %e A354488 tan(phi) = 3/9 = 1/3, tan(2*phi) = 2*(1/3)/(1 - (1/3)^2) = (2/3)/(8/9) = 18/24 = 3/4. %e A354488 . %e A354488 3 | . . . . . C . . . . 3 | . . . . . C . . . . %e A354488 2 | . . . . . . . . . . 2 | D . . . . . . . . B %e A354488 1 | D . . . . . . . . B 1 | . . . . . . . . . . %e A354488 0 | . A . . . . . . . . 0 | . A . . . . . . . . %e A354488 y /-------------------- y /-------------------- %e A354488 x 0 1 2 3 4 5 6 7 8 9 x 0 1 2 3 4 5 6 7 8 9 %e A354488 . %e A354488 3 | . . . . . . C . . . 3 | . . . . . . C . . . %e A354488 2 | . . . . . . . . . . 2 | D . . . . . . . . B %e A354488 1 | D . . . . . . . . B 1 | . . . . . . . . . . %e A354488 0 | . . A . . . . . . . 0 | . . A . . . . . . . %e A354488 y /-------------------- y /-------------------- %e A354488 x 0 1 2 3 4 5 6 7 8 9 x 0 1 2 3 4 5 6 7 8 9 %e A354488 . %e A354488 T(12,6) = 3, with slopes of diagonals of quadrilateral against y = 0: sAC, sDB, sAC = 6/2 = 3, sDB = 4/12 = 1/3, angle difference theta = sAC - sDB. %e A354488 Using tan(alpha - beta) = (tan(alpha) - tan(beta))/(1 + tan(alpha)*tan(beta)), tan(theta) = (sAC - sBD) / (1 + sAC*sBD) = (3 - 1/3)/( 1 + 1 ) = 4/3. %e A354488 tan(phi) = 6/12 = 1/2; tan(2*phi) = 2*(1/2)/(1 - (1/2)^2) = 1/(3/4) = 4/3. %e A354488 . %e A354488 6 | . . . C . . . . . . . . . 6 | . . . . C . . . . . . . . %e A354488 5 | . . . . . . . . . . . . B 5 | . . . . . . . . . . . . B %e A354488 4 | . . . . . . . . . . . . . 4 | . . . . . . . . . . . . . %e A354488 3 | . . . . . . . . . . . . . 3 | . . . . . . . . . . . . . %e A354488 2 | . . . . . . . . . . . . . 2 | . . . . . . . . . . . . . %e A354488 1 | D . . . . . . . . . . . . 1 | D . . . . . . . . . . . . %e A354488 0 | . A . . . . . . . . . . . 0 | . . A . . . . . . . . . . %e A354488 y /-------------------------- y /-------------------------- %e A354488 x 0 1 2 3 4 5 6 7 8 9 0 1 2 x 0 1 2 3 4 5 6 7 8 9 0 1 2 %e A354488 . %e A354488 6 | . . . . . . C . . . . . . %e A354488 5 | . . . . . . . . . . . . B %e A354488 4 | . . . . . . . . . . . . . %e A354488 3 | . . . . . . . . . . . . . %e A354488 2 | . . . . . . . . . . . . . %e A354488 1 | D . . . . . . . . . . . . %e A354488 0 | . . . . A . . . . . . . . %e A354488 y /-------------------------- %e A354488 x 0 1 2 3 4 5 6 7 8 9 0 1 2 %o A354488 (PARI) \\ See link. The program a354488(w1,w2) prints a list of the nonzero terms [w, d, T_a353532(w+1,d+1), T(w,d)] in the range w1 <= w <= w2. %Y A354488 Cf. A353532, A353533. %Y A354488 A354489 provides the widths of those grid rectangles for which no inserted quadrilaterals with matching angles of the diagonals exist, i.e., all terms = 0 in a row of the triangle. %K A354488 nonn,tabl %O A354488 4,12 %A A354488 _Hugo Pfoertner_ and _Rainer Rosenthal_, May 28 2022