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

5,2

Links

Table of n, a(n) for n=5..79.

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
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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
.
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
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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

Cf. A353447, A353532.

Sequence in context: A031265 A029202 A173894 * A349041 A285870 A224931

Adjacent sequences: A353530 A353531 A353532 * A353534 A353535 A353536

Keyword

nonn,tabl

Author

Hugo Pfoertner and Rainer Rosenthal, May 04 2022

Status

approved