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%I #9 May 29 2022 17:59:35
%S 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 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 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 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 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 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 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 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 Hugo Pfoertner, <a href="/A354488/a354488.gp.txt">PARI program to print list of nonzero sequence terms</a>.
%e The triangle begins:
%e 4: 0,
%e 5: 0,0,
%e 6: 0,0, 0,
%e 7: 0,0, 0, 0,
%e 8: 0,3, 0, 0, 0,
%e 9: 4,0, 0, 0, 0, 0,
%e 10: 0,0, 0, 0, 0, 0, 0,
%e 11: 0,0, 0, 0, 0, 0, 0, 0,
%e 12: 0,0, 0, 3, 0,11, 0, 0,0,
%e 13: 0,0, 0, 0, 0, 0, 0, 0,0, 0,
%e 14: 0,0, 0, 0,12, 0, 0, 0,0, 0,0,
%e 15: 0,0, 0, 0, 0, 0, 0,32,0, 0,0, 0,
%e 16: 0,0, 0, 0, 0,23, 0, 0,0, 0,0, 0, 0,
%e 17: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,
%e 18: 0,0, 0,33, 0, 0,51, 0,0, 53,0, 0, 0, 0,0,
%e 19: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,
%e 20: 0,0, 0, 0, 0, 0, 0, 0,0, 46,0, 0, 0, 0,0, 0,0,
%e 21: 0,0, 0, 0,18, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0,
%e 22: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0, 0,
%e 23: 0,0, 0, 0, 0, 0, 0, 0,0, 0,0, 0, 0, 0,0, 0,0,0, 0,0,
%e 24: 0,0, 0, 0, 0,53, 0, 0,0,107,0, 0, 0,57,0,91,0,0, 0,0,0,
%e 25: 0,0,24, 0, 0, 0, 0, 0,0, 0,0, 0,108, 0,0, 0,0,0, 0,0,0,0,
%e 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 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 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 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 n ------------------------------------------------------------------
%e m: 3 4 5 6 7 8 9 10 . 12 . 14 15 16 . 18 . . 21 . . . . . . 28
%e .
%e T(8,4) = 3, tan(theta) = 4/3 = tan(2*phi).
%e Intersection angle of diagonals of the grid rectangle:
%e tan(2*phi) = 2 *(1/2) / (1 - (1/2)^2) = 1 / (3/4) = 4/3, with tan(phi) = 4/8 = 1/2.
%e .
%e 4 | . . . . . C . . . 4 | . . . . . C . . . 4 | . . . . . . C . .
%e 3 | . . . . . . . . . 3 | . . . . . . . . . 3 | . . . . . . . . .
%e 2 | . . . . . . . . . 3 | D . . . . . . . B 2 | . . . . . . . . .
%e 1 | D . . . . . . . B 1 | . . . . . . . . . 1 | D . . . . . . . B
%e 0 | . . A . . . . . . 0 | . . A . . . . . . 0 | . . . A . . . . .
%e y /------------------ y /------------------ y /------------------
%e 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 .
%e T(9,3) = 4, tan(theta) = 3/4 = tan(2*phi).
%e 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 .
%e 3 | . . . . . C . . . . 3 | . . . . . C . . . .
%e 2 | . . . . . . . . . . 2 | D . . . . . . . . B
%e 1 | D . . . . . . . . B 1 | . . . . . . . . . .
%e 0 | . A . . . . . . . . 0 | . A . . . . . . . .
%e y /-------------------- y /--------------------
%e x 0 1 2 3 4 5 6 7 8 9 x 0 1 2 3 4 5 6 7 8 9
%e .
%e 3 | . . . . . . C . . . 3 | . . . . . . C . . .
%e 2 | . . . . . . . . . . 2 | D . . . . . . . . B
%e 1 | D . . . . . . . . B 1 | . . . . . . . . . .
%e 0 | . . A . . . . . . . 0 | . . A . . . . . . .
%e y /-------------------- y /--------------------
%e x 0 1 2 3 4 5 6 7 8 9 x 0 1 2 3 4 5 6 7 8 9
%e .
%e 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 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 tan(phi) = 6/12 = 1/2; tan(2*phi) = 2*(1/2)/(1 - (1/2)^2) = 1/(3/4) = 4/3.
%e .
%e 6 | . . . C . . . . . . . . . 6 | . . . . C . . . . . . . .
%e 5 | . . . . . . . . . . . . B 5 | . . . . . . . . . . . . B
%e 4 | . . . . . . . . . . . . . 4 | . . . . . . . . . . . . .
%e 3 | . . . . . . . . . . . . . 3 | . . . . . . . . . . . . .
%e 2 | . . . . . . . . . . . . . 2 | . . . . . . . . . . . . .
%e 1 | D . . . . . . . . . . . . 1 | D . . . . . . . . . . . .
%e 0 | . A . . . . . . . . . . . 0 | . . A . . . . . . . . . .
%e y /-------------------------- y /--------------------------
%e 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 .
%e 6 | . . . . . . C . . . . . .
%e 5 | . . . . . . . . . . . . B
%e 4 | . . . . . . . . . . . . .
%e 3 | . . . . . . . . . . . . .
%e 2 | . . . . . . . . . . . . .
%e 1 | D . . . . . . . . . . . .
%e 0 | . . . . A . . . . . . . .
%e y /--------------------------
%e x 0 1 2 3 4 5 6 7 8 9 0 1 2
%o (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 Cf. A353532, A353533.
%Y 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 nonn,tabl
%O 4,12
%A _Hugo Pfoertner_ and _Rainer Rosenthal_, May 28 2022