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A353532 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. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
3,5
COMMENTS
T(n,m) is a triangle, read by rows.
LINKS
Rainer Rosenthal, Rows n = 3..100, flattened
Hugo Pfoertner, PARI program
EXAMPLE
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
.
.
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
.
.
T(5,4) = a(5) = 3:
.
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
.
10-13-5-2 13-10-5-2 13-5-8-2
9 X 16 9 X 17 10 X 17
.
T(5,5) = a(6) = A353447(5) = 1:
.
5 | . . . X .
4 | . . . . .
3 | . . . . X 13-5-18-2
2 | X . . . . 20 X 17
1 | . X . . .
y /----------
x 1 2 3 4 5
.
T(6,3) = a(7) = 1:
.
3 | . . . X . .
2 | X . . . . X 17-5-10-2
1 | . X . . . . 8 X 25
y /------------
x 1 2 3 4 5 6
.
T(6,4) = a(8) = 7:
.
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
.
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
.
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
.
17-5-20-2 10-13-8-5 13-10-8-5
18 X 25 9 X 25 9 X 26
.
PROG
(PARI) see Pfoertner link.
CROSSREFS
The general case without excluding the corners of the grid rectangle is covered in A354700 and A354701.
Sequence in context: A137470 A112492 A210574 * A049290 A297191 A147990
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)