OFFSET
5,2
EXAMPLE
The quadrilaterals counted in A353532 with m = 3 or m = n cannot have perpendicular diagonals, and are therefore omitted in the triangle of this sequence.
.
\ m 3 4 5 6 7 8 9 10 11
n \-----------------------------------
3 | 0, | | | | | | | |
4 | 0, 0, | | | | | | |
5 | 0, 1, 0, | | | | | |
6 | 0, 2, 1, 0, | | | | |
7 | 0, 2, 2, 3, 0, | | | |
8 | 0, 3, 3, 4, 6, 0, | | |
9 | 0, 3, 5, 5, 8, 9, 0, | |
10 | 0, 4, 4, 6, 12, 12, 12, 0, |
11 | 0, 4, 4, 12, 8, 11, 15, 14, 0
.
T(5,4) = a(1) = 1:
.
4 | . C . . . Squared distances denoted
3 | . . . . . as in examples A353532:
2 | D . . . B
1 | . A . . . AB-BC-CD-DA (around)
y /---------- AC X DB (across)
x 1 2 3 4 5
.
10-13-5-2
9 X 16
.
T(6,4) = a(2) = 2:
.
4 | . X . . . . 4 | . . X . . .
3 | . . . . . . 3 | . . . . . .
2 | X . . . . X 2 | X . . . . X
1 | . X . . . . 1 | . . X . . .
y /------------ y /------------
x 1 2 3 4 5 6 x 1 2 3 4 5 6
.
17-20-5-2 10-13-8-5
9 X 25 9 X 25
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T(6,5) = a(3) = 1:
.
5 | . . X . . .
4 | . . . . . .
3 | . . . . . . 10-18-13-5
2 | X . . . . X 16 X 25
1 | . . X . . .
y /------------
x 1 2 3 4 5 6
.
T(9,5) = a(12) = 5;
3 quadrilaterals with diagonals parallel to the grid axes:
.
5 | . X . . . . . . . 5 | . . X . . . . . . 5 | . . . X . . . . .
4 | . . . . . . . . . 4 | . . . . . . . . . 4 | . . . . . . . . .
3 | . . . . . . . . . 3 | . . . . . . . . . 3 | . . . . . . . . .
2 | X . . . . . . . X 2 | X . . . . . . . X 2 | X . . . . . . . X
1 | . X . . . . . . . 1 | . . X . . . . . . 1 | . . . X . . . . .
y /------------------ y /------------------ y /------------------
x 1 2 3 4 5 6 7 8 9 x 1 2 3 4 5 6 7 8 9 x 1 2 3 4 5 6 7 8 9
.
50-58-10-2 37-45-13-5 26-34-18-10
16 X 64 16 X 64 16 X 64
.
The 2 quadrilaterals with diagonals not aligned with the grid axes are the smallest example of this type:
.
.
5 | . X . . . . . . . 5 | . . X . . . . . .
4 | . . . . . . . . X 4 | . . . . . . . . X
3 | . . . . . . . . . 3 | . . . . . . . . .
2 | X . . . . . . . . 2 | X . . . . . . . .
1 | . . X . . . . . . 1 | . . . X . . . . .
y /------------------ y /------------------
x 1 2 3 4 5 6 7 8 9 x 1 2 3 4 5 6 7 8 9
.
45-50-10-5 34-37-13-10
17 X 68 17 X 68
.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner and Rainer Rosenthal, May 04 2022
STATUS
approved