Catalogue of arrangements for A337515

Arrangements of n squares with side lengths s = 1, 2, ..., n
such that the number of 1 X 1 cells formed by the boundaries of all squares
is maximized.

Exhaustive enumeration by Hugo Pfoertner in September 2020.

Number of essentially distinct solutions: A337515(n).
Number of 1X1 cells: A336660(n).

The position of a square is given by the coordinate pair (x,y) of the
lower left vertex. Squares are noted in descending order of size.
Arrangements which differ only in the position (x1,y1) are called 1-siblings.
Coordinates of squares n, n-1, ..., 2 are given only for the first 1-sibling.

This list serves as a catalogue for the illustrations.
Only the first 1-sibling is shown in full. Each position (x1,y1) is marked
by a little cross (+).

n = 1, 1 solutions, 1X1 cells: 1

  x1 y1
   0  0


n = 2, 1 solutions, 1X1 cells: 1

  x2 y2  x1 y1
   0  0   0  0


n = 3, 1 solutions, 1X1 cells: 4

  x3 y3  x2 y2  x1 y1
   0  0   2  2   3  3


n = 4, 4 solutions, 1X1 cells: 7

  x4 y4  x3 y3  x2 y2  x1 y1
   0  0   2  2   1  3   3  2
   0  0   2  3   1  3   0  2
  .. ..  .. ..  .. ..   3  5
  .. ..  .. ..  .. ..   4  4


n = 5, 1 solutions, 1X1 cells: 12

  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
   0  0   3  4   1  3   0  4   2  3


n = 6, 2 solutions, 1X1 cells: 17

  x6 y6  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
   0  0   4  4   1  3   4  2   5  1   2  6
   0  0   4  4   3  5   5  3   7  3   3  7


n = 7, 1 solutions, 1X1 cells: 24

  x7 y7  x6 y6  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
   0  0   0  2   5  4   5  1   5  3   6  0   8  2


n = 8, 8 solutions, 1X1 cells: 31

  x8 y8  x7 y7  x6 y6  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
   0  0   0  2   4  6   5  5   2  7   6  4   1  7   4 10
   0  0   2  3   7  0   5  2   7  5   6  1   8  8  10  7
   0  0   3  7   0  3   2  6   5  5   1  7   7  4   3  6
  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   8  8
   0  0   3  7   0  3   2  6   5  6   1  7   6  9   4 10
  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   8  8
   0  0   4  6   1  3   5  5   2  7   6  4   3 10   2  9
   0  0   4  6   5  5   2  4   6  3   3  7   7  2   9  4


n = 9, 4 solutions, 1X1 cells: 42

  x9 y9  x8 y8  x7 y7  x6 y6  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
   0  0   5  7   3  4   6  6   2  5   4  8   8  5   7 10   9  9
   0  0   5  7   4  5   2  4   7  6   6  4   3  8   5  4   4  6
   0  0   6  6   3  4   5  7   2  5   4  8   8  5   7 10   9  9
   0  0   6  6   4  5   2  4   5  8   3  7   7  4   3  6   5  5


n = 10, 21 solutions, 1X1 cells: 50

  x10 y10  x9 y9  x8 y8  x7 y7  x6 y6  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
    2   0   0  9   6  5   3  8   5  6   7  9   4  7   8 11   7  7  11 11
    2   0   0  9   8  6   4  4   7  7   9  4   6  8  10  5   3  9   9  4
  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  13  4
  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  13  7
    0   0   0  2   5  8   4  7   2  6   1  8   3  9   7  6   1  7   1 12
  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   3  6
  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   5  7
    0   0   0  3   8  7   8  1   7  5   7  4   7  2  11  6   9  1   7  7
    0   0   0  7   7  6   6  5   5  3   9  2   8  4   5  8   8 11   6  5
  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   7 13
    0   0   0  8   6  7   4  4   7  6   9  3   8  5   5  9   3  7  10 10
    0   0   2  7   8  3   7  5   6  8  10 10   9  9   6  6   9  4   7 11
  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   8  3
    0   0   2  8   7  3   5  7   8  6  10 10   9  9   6  6   4  9  11  7
    0   0   3  3   7  9   4  6   8  8  10 10   9  7   6 11   3 11   5 10
    0   0   3  8   7  3   4  7   8  6  10 10   9  9   6  6   3  7   5  9
    0   0   3  9   4  8   1  6   1  6   6  7   5  7   2 11   0  9   6 12
    1   0   5  4   0  9   2  8   4  6   6  9   3  7   7 11   6  7  10 11
    0   0   5  7   7  1   6  6   3  5   2  9   4  8   8  5  12  6  11  8
    0   0   6  7   1  8   4  4   7  6   9  3   8  5   5  9   3  7  10 10
    0   0   6  7   7  1   5  6   3  5   2  9   4  8   8  5  11  8   5  6


n = 11, 29 solutions, 1X1 cells: 65

  x11 y11  x10 y10  x9 y9  x8 y8  x7 y7  x6 y6  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
    0   0    2   4   7  9   8  8   6  5   9  7   5 10  10  6   6 13  11 11   5 12
    0   0    2   4   8  8   7  9   6  5   9  7   5 10  10  6   6 13  11 11   5 12
    0   0    5   7   8  8   2  4   7  6   6  4   4  9   9  5   3 10   6  6   6 13
    0   0    5   8   0  3   6  7   1  6   4  9   7  5   3 10  10  6   8  4   0  9
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   1 11
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   1 12
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   2 11
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   2 12
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   5  6
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   7  3
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   7 14
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   9 12
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  10  3
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  13  9
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  14  9
    0   0    6   6   7  9   2  4   5  8   3  7   8  5   4 10   9  4   5  7  12  7
    0   0    6   7   0  3   5  8   1  6   4  9   7  5   3 10  10  6   8  4   0  9
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   1 11
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   1 12
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   2 11
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   2 12
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   5  6
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   7  3
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   7 14
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   7 15
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   7 16
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   9 12
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  10  3
  ... ...  ... ...  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  12 10