OFFSET
4,12
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).
LINKS
EXAMPLE
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
PROG
(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.
CROSSREFS
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.
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner and Rainer Rosenthal, May 28 2022
STATUS
approved