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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.
2
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, 12, 30, 18, 27, 8, 4, 11, 11, 12, 24, 25, 33, 41, 18, 10, 17, 21, 17, 36, 24, 35, 32, 38, 0, 8, 17, 19, 21, 51, 43, 65, 84, 87, 57, 62, 15, 24, 31, 25, 49, 31, 48, 45, 53, 33, 76, 0
OFFSET
2,8
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.
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.
EXAMPLE
The triangle begins, with corresponding terms of A353532 shown in parenthesis:
\ d 2 3 4 5 6 7 8 9
w \---------------------------------------------------------------------
2 | 0 ( 0) | | | | | | |
3 | 0 ( 0) 0 ( 0) | | | | | |
4 | 0 ( 0) 1 ( 3) 0 ( 1) | | | | |
5 | 1 ( 1) 3 ( 7) 1 ( 12) 0 ( 11) | | | |
6 | 0 ( 1) 3 (11) 3 ( 26) 4 ( 52) 4 ( 40) | | |
7 | 3 ( 4) 6 (23) 6 ( 50) 6 ( 94) 12 (147) 0 (105) | |
8 | 2 ( 4) 6 (30) 7 ( 69) 9 (127) 15 (198) 13 (301) 6 (190) |
9 | 6 (10) 10 (49) 12 (103) 12 (192) 30 (302) 18 (444) 27 (583) 8 (379)
.
Only 1 = T(4,3) of the 3 = T_a353532(5,4) quadrilaterals has diagonals AC, BD whose intersection point S has integer coordinates:
.
3 | . C . . . 3 | . C . . . 3 | . . C . .
2 | . . . . . 2 | . . . . B 2 | . . . . B
1 | D S . . B 1 | D . . . . 1 | D . . . .
0 | . A . . . 0 | . A . . . 0 | . A . . .
y /---------- y /---------- y /----------
x 0 1 2 3 4 x 0 1 2 3 4 x 0 1 2 3 4
S=(1,1) S=(1,5/4) S=(16/11,15/11)
.
T(5,2) = T_a353532(6,3) = 1:
.
2 | . . . C . .
1 | D . S . . B
0 | . A . . . .
y /------------
x 0 1 2 3 4 5
S=(2,1)
.
T(5,3) = 3 of the T_a353532(6,4) = 7 intersection points S of the diagonals AC, BD have integer coordinates:
.
3 | . C . . . . 3 | . C . . . . 3 | . . C . . . 3 | . . . C . .
2 | . . . . . . 2 | . . . . . B 2 | . . . . . . 2 | D . . . . .
1 | D S . . . B 1 | D . . . . . 1 | D . . . . B 1 | . . . . . B
0 | . A . . . . 0 | . A . . . . 0 | . A . . . . 0 | . A . . . .
y /------------ y /------------ y /------------ y /------------
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
S=(1,1) S=(1,6/5) S=(4/3,1) S=(35/17,27/17)
.
3 | . . . . C . 3 | . . C . . . 3 | . . C . . .
2 | . . . . . . 2 | . . . . . . 2 | . . . . . B
1 | D . S . . B 1 | D . S . . B 1 | D . . . . .
0 | . A . . . . 0 | . . A . . . 0 | . . A . . .
y /------------ y /------------ y /------------
x 0 1 2 3 4 5 x 0 1 2 3 4 5 x 0 1 2 3 4 5
S=(2,1) S=(2,1) S=(2,7/5)
PROG
(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.
CROSSREFS
A354491 is the diagonal of the triangle.
Sequence in context: A085604 A345371 A306268 * A365970 A144357 A122848
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner, May 30 2022
STATUS
approved