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A354490
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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.
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2
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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
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OFFSET
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2,8
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=2..79.
Hugo Pfoertner, PARI program to print sequence terms.
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EXAMPLE
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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)
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Only 1 = T(4,3) of the 3 = T_a353532(5,4) quadrilaterals has diagonals AC, BD whose intersection point S has integer coordinates:
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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)
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T(5,2) = T_a353532(6,3) = 1:
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2 | . . . C . .
1 | D . S . . B
0 | . A . . . .
y /------------
x 0 1 2 3 4 5
S=(2,1)
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T(5,3) = 3 of the T_a353532(6,4) = 7 intersection points S of the diagonals AC, BD have integer coordinates:
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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)
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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)
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PROG
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(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.
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CROSSREFS
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Cf. A353532, A353533, A354488.
A354491 is the diagonal of the triangle.
Sequence in context: A085604 A345371 A306268 * A144357 A122848 A272481
Adjacent sequences: A354487 A354488 A354489 * A354491 A354492 A354493
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KEYWORD
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nonn,tabl
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AUTHOR
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Hugo Pfoertner, May 30 2022
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STATUS
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approved
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