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A353532
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T(n,m) is the number of non-congruent quadrilaterals with integer vertex coordinates (x1,1), (n,y2), (x3,m), (1,y4), 1 < x1, x3 < n, 1 < y2, y4 < m, m <= n, such that the 6 distances between the 4 vertices are distinct.
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13
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0, 0, 0, 0, 3, 1, 1, 7, 12, 11, 1, 11, 26, 52, 40, 4, 23, 50, 94, 147, 105, 4, 30, 69, 127, 198, 301, 190, 10, 49, 103, 192, 302, 444, 583, 379, 10, 58, 127, 244, 387, 576, 754, 1039, 616, 18, 84, 180, 329, 509, 756, 989, 1334, 1680, 987, 18, 94, 209, 389, 611, 910, 1203, 1618, 2052, 2581, 1426
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OFFSET
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3,5
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COMMENTS
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T(n,m) is a triangle, read by rows.
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LINKS
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EXAMPLE
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The triangle begins
\ m 3 4 5 6 7 8 9 10
n \-------------------------------------
3 | 0, | | | | | | |
4 | 0, 0, | | | | | |
5 | 0, 3, 1, | | | | |
6 | 1, 7, 12, 11, | | | |
7 | 1, 11, 26, 52, 40, | | |
8 | 4, 23, 50, 94, 147, 105, | |
9 | 4, 30, 69, 127, 198, 301, 190, |
10 | 10, 49, 103, 192, 302, 444, 583, 379
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4 | . C . . . There are six squared distances.
3 | . . . . . They are arranged as follows:
2 | D . . . B AB-BC-CD-DA (counterclockwise)
1 | . A . . . AC X DB (across)
y /---------- Here: AB = 3^2 + 1^2 = 10,
x 1 2 3 4 5 BC = 13, CD = 5, DA = 2,
. AC = 9, DB = 16
10-13-5-2 <==== yielding this
9 X 16 <==== description
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T(5,4) = a(5) = 3:
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4 | . X . . . 4 | . X . . . 4 | . . X . .
3 | . . . . . 3 | . . . . X 3 | . . . . X
2 | X . . . X 2 | X . . . . 2 | X . . . .
1 | . X . . . 1 | . X . . . 1 | . X . . .
y /---------- y /---------- y /----------
x 1 2 3 4 5 x 1 2 3 4 5 x 1 2 3 4 5
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10-13-5-2 13-10-5-2 13-5-8-2
9 X 16 9 X 17 10 X 17
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5 | . . . X .
4 | . . . . .
3 | . . . . X 13-5-18-2
2 | X . . . . 20 X 17
1 | . X . . .
y /----------
x 1 2 3 4 5
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T(6,3) = a(7) = 1:
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3 | . . . X . .
2 | X . . . . X 17-5-10-2
1 | . X . . . . 8 X 25
y /------------
x 1 2 3 4 5 6
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T(6,4) = a(8) = 7:
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4 | . X . . . . 4 | . X . . . . 4 | . . X . . . 4 | . . . X . .
3 | . . . . . . 3 | . . . . . X 3 | . . . . . . 3 | X . . . . .
2 | X . . . . X 2 | X . . . . . 2 | X . . . . X 2 | . . . . . X
1 | . X . . . . 1 | . X . . . . 1 | . X . . . . 1 | . X . . . .
y /------------ y /------------ y /------------ y /------------
x 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6
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17-20-5-2 20-17-5-2 17-13-8-2 17-8-10-5
9 X 25 9 X 26 10 X 25 13 X 26
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4 | . . . . X . 4 | . . X . . . 4 | . . X . . .
3 | . . . . . . 3 | . . . . . . 3 | . . . . . X
2 | X . . . . X 2 | X . . . . X 2 | X . . . . .
1 | . X . . . . 1 | . . X . . . 1 | . . X . . .
y /------------ y /------------ y /------------
x 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6
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17-5-20-2 10-13-8-5 13-10-8-5
18 X 25 9 X 25 9 X 26
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PROG
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(PARI) see Pfoertner link.
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CROSSREFS
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The general case without excluding the corners of the grid rectangle is covered in A354700 and A354701.
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KEYWORD
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AUTHOR
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STATUS
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approved
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