A353533 T(n,m) with 4 <= m < n is the number of quadrilaterals in A353532 with perpendicular diagonals, where T(n,m) is a triangle read by rows.
1, 2, 1, 2, 2, 3, 3, 3, 4, 6, 3, 5, 5, 8, 9, 4, 4, 6, 12, 12, 12, 4, 4, 12, 8, 11, 15, 14, 5, 5, 8, 10, 15, 15, 20, 18, 5, 5, 8, 27, 15, 33, 32, 26, 25, 6, 6, 10, 11, 17, 17, 23, 22, 29, 29, 6, 6, 10, 12, 48, 18, 24, 29, 30, 42, 34, 7, 7, 16, 14, 21, 21, 41, 69, 34
Offset: 5
Examples
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.
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\ 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
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T(5,4) = a(1) = 1:
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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
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10-13-5-2
9 X 16
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T(6,4) = a(2) = 2:
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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
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17-20-5-2 10-13-8-5
9 X 25 9 X 25
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T(6,5) = a(3) = 1:
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5 | . . X . . .
4 | . . . . . .
3 | . . . . . . 10-18-13-5
2 | X . . . . X 16 X 25
1 | . . X . . .
y /------------
x 1 2 3 4 5 6
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T(9,5) = a(12) = 5;
3 quadrilaterals with diagonals parallel to the grid axes:
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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
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50-58-10-2 37-45-13-5 26-34-18-10
16 X 64 16 X 64 16 X 64
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The 2 quadrilaterals with diagonals not aligned with the grid axes are the smallest example of this type:
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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
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45-50-10-5 34-37-13-10
17 X 68 17 X 68
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