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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.

Original entry on oeis.org

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, 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, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0
Offset: 4

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Author

Hugo Pfoertner and Rainer Rosenthal, May 28 2022

Keywords

Comments

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.
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.
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.
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).

Examples

			The triangle begins:
   4: 0,
   5: 0,0,
   6: 0,0, 0,
   7: 0,0, 0, 0,
   8: 0,3, 0, 0, 0,
   9: 4,0, 0, 0, 0, 0,
  10: 0,0, 0, 0, 0, 0, 0,
  11: 0,0, 0, 0, 0, 0, 0, 0,
  12: 0,0, 0, 3, 0,11, 0, 0,0,
  13: 0,0, 0, 0, 0, 0, 0, 0,0,  0,
  14: 0,0, 0, 0,12, 0, 0, 0,0,  0,0,
  15: 0,0, 0, 0, 0, 0, 0,32,0,  0,0, 0,
  16: 0,0, 0, 0, 0,23, 0, 0,0,  0,0, 0,  0,
  17: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,
  18: 0,0, 0,33, 0, 0,51, 0,0, 53,0, 0,  0, 0,0,
  19: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,
  20: 0,0, 0, 0, 0, 0, 0, 0,0, 46,0, 0,  0, 0,0, 0,0,
  21: 0,0, 0, 0,18, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,
  22: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,
  23: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,0,
  24: 0,0, 0, 0, 0,53, 0, 0,0,107,0, 0,  0,57,0,91,0,0,  0,0,0,
  25: 0,0,24, 0, 0, 0, 0, 0,0,  0,0, 0,108, 0,0, 0,0,0,  0,0,0,0,
  26: 0,0, 0, 0, 0, 0, 0, 0,0,  0,0, 0,  0, 0,0, 0,0,0,  0,0,0,0,0,
  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,
  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,
  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
   n  ------------------------------------------------------------------
   m: 3 4  5  6  7  8  9 10 .  12 . 14  15 16 . 18 . .  21 . . . . . . 28
.
T(8,4) = 3, tan(theta) = 4/3 = tan(2*phi).
Intersection angle of diagonals of the grid rectangle:
tan(2*phi) = 2 *(1/2) / (1 - (1/2)^2) = 1 / (3/4) = 4/3, with tan(phi) = 4/8 = 1/2.
.
  4 | . . . . . C . . .   4 | . . . . . C . . .   4 | . . . . . . C . .
  3 | . . . . . . . . .   3 | . . . . . . . . .   3 | . . . . . . . . .
  2 | . . . . . . . . .   3 | D . . . . . . . B   2 | . . . . . . . . .
  1 | D . . . . . . . B   1 | . . . . . . . . .   1 | D . . . . . . . B
  0 | . . A . . . . . .   0 | . . A . . . . . .   0 | . . . A . . . . .
  y /------------------   y /------------------   y /------------------
    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
.
T(9,3) = 4, tan(theta) = 3/4 = tan(2*phi).
tan(phi) = 3/9 = 1/3, tan(2*phi) = 2*(1/3)/(1 - (1/3)^2) = (2/3)/(8/9) = 18/24 = 3/4.
.
  3 | . . . . . C . . . .       3 | . . . . . C . . . .
  2 | . . . . . . . . . .       2 | D . . . . . . . . B
  1 | D . . . . . . . . B       1 | . . . . . . . . . .
  0 | . A . . . . . . . .       0 | . A . . . . . . . .
  y /--------------------       y /--------------------
    x 0 1 2 3 4 5 6 7 8 9         x 0 1 2 3 4 5 6 7 8 9
.
  3 | . . . . . . C . . .       3 | . . . . . . C . . .
  2 | . . . . . . . . . .       2 | D . . . . . . . . B
  1 | D . . . . . . . . B       1 | . . . . . . . . . .
  0 | . . A . . . . . . .       0 | . . A . . . . . . .
  y /--------------------       y /--------------------
    x 0 1 2 3 4 5 6 7 8 9         x 0 1 2 3 4 5 6 7 8 9
.
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.
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.
tan(phi) = 6/12 = 1/2; tan(2*phi) = 2*(1/2)/(1 - (1/2)^2) = 1/(3/4) = 4/3.
.
  6 | . . . C . . . . . . . . .       6 | . . . . C . . . . . . . .
  5 | . . . . . . . . . . . . B       5 | . . . . . . . . . . . . B
  4 | . . . . . . . . . . . . .       4 | . . . . . . . . . . . . .
  3 | . . . . . . . . . . . . .       3 | . . . . . . . . . . . . .
  2 | . . . . . . . . . . . . .       2 | . . . . . . . . . . . . .
  1 | D . . . . . . . . . . . .       1 | D . . . . . . . . . . . .
  0 | . A . . . . . . . . . . .       0 | . . A . . . . . . . . . .
  y /--------------------------       y /--------------------------
    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
.
  6 | . . . . . . C . . . . . .
  5 | . . . . . . . . . . . . B
  4 | . . . . . . . . . . . . .
  3 | . . . . . . . . . . . . .
  2 | . . . . . . . . . . . . .
  1 | D . . . . . . . . . . . .
  0 | . . . . A . . . . . . . .
  y /--------------------------
    x 0 1 2 3 4 5 6 7 8 9 0 1 2

Links

Crossrefs

Cf. A353532, A353533.
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.

Programs

  • 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.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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