A335232 -id:A335232 - cp's OEIS frontend

A271490 Size of maximal subset of points of n X n grid such that no two points are at the same distance.

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 13, 13

Offset: 1

N. J. A. Sloane, Apr 14 2016

Inverse function to A193838, which is the main entry for this problem.

			From _Ehit Dinesh Agarwal_, May 28 2020: (Start)
An 11 X 11 grid has only two subsets of size 10, barring symmetry: {(0,0), (0,2), (0,3), (0,7), (1,10), (5,4), (6,0), (8,7), (9,8), (10, 10)} and {(0,0), (0,6), (0,7), (1,2), (4,10), (7,8), (7,10), (9,2), (9,6), (10,5)}.
A 12 x 13 grid has only four subsets of size 11, barring symmetry: {(0,0), (0,1), (0,9), (0,12), (2,0), (5,3), (6,12), (7,0), (8,4), (10,10), (11,11)}. (End)

R. K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer New York, 2004, F2, 367-368.
Keith F. Lynch, Posting to Math Fun Mailing List, Apr 02 2016.

P. Erdős and R. K. Guy, Distinct distances between lattice points, Elemente der Mathematik 25 (1970), 121-123.
Wolfram Demonstration Project, No Repeated Distances
A. Zimmermann, Al Zimmermann's Programming Contests: Point Packing. (Oct 10, 2009).

Cf. A193838, A335232 (number of solutions).

a(11)-a(13) corrected and extended by Ehit Dinesh Agarwal, May 28 2020
a(14)-a(16) from Bert Dobbelaere, Sep 20 2020
a(17) from Fausto A. C. Cariboni, Jul 16 2022

A351699 T(n,k) is the number of non-congruent maximal subsets of a grid of n X k lattice points (k <= n), such that no two points are at the same distance, and the points do not fit into a smaller grid. The size of the subsets is given by A351700. T(n,k) and A351700 are triangles read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 5, 10, 1, 5, 28, 7, 21, 2, 19, 8, 104, 330, 2, 1, 4, 70, 15, 110, 574, 1, 3, 30, 272, 205, 4, 71, 563, 1991, 4, 68, 50, 1001, 113, 1130, 4, 76, 383, 9, 8, 362, 35, 1150, 23, 363, 3975, 7, 38, 8, 18, 1082, 415, 2, 638, 7503, 23, 515, 5802, 2, 2, 150, 62, 4238, 120, 1, 55, 1776, 17277, 26, 481, 2388

Offset: 1

Rainer Rosenthal and Hugo Pfoertner, Apr 09 2022

Configurations of points differing by any combination of rotation and reflection are counted only once.

			The triangle begins:
  #
  # 1:  1                   Counting grids n X k.
      ( 1 )                 Two lines per side length n:
  # 2:  2  2                1. for other side k = 1, 2, ...
      ( 1  1 )                 maximal number of points
  # 3:  2  3    3           2. number of configurations
      ( 1  2    1 )
  # 4:  3  4    4    4      Example: 28 figures with
      ( 1  1    5   10 )             4 points on 5 X 3
  # 5:  3  4    4    5    5
      ( 1  5   28    7   21 )
  # 6:  3  4    5    5    5    6
      ( 2 19    8  104  330    2 )
  # 7:  4  5    5    6    6    6    7
      ( 1  4   70   15  110  574    1 )
  # 8:  4  5    5    6    7    7    7     7
      ( 3 30  272  205    4   71  563  1991 )
  # 9:  4  5    6    6    7    7    8    8   8
      ( 4 68   50 1001  113 1130    4   76 383 )
  #10:  4  6    6    7    7    8    8    8   9    9
      ( 9  8  362   35 1150   23  363 3975   7   38 )
  #11:  4  6    6    7    8    8    8    9   9    9 10
      ( 8 18 1082  415    2  638 7503   23 515 5802  2 )
  #
  #   Grid n X k configurations with
  #       distinct distances
  .
  .
  All T(6,3) = 8 configurations
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  X  X  .  X  .               2 |  .  .  .  .  X  .
      1 |  .  .  .  .  .  X               1 |  .  .  .  .  .  X
      0 |  X  .  .  .  .  .               0 |  X  .  X  .  .  X
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,2,4,5,8,9,10,17,20,26}  dist^2   {1,2,4,5,8,9,10,20,25,26}
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  .  X  .  X  .               2 |  .  X  .  X  .  .
      1 |  .  .  .  .  .  X               1 |  X  .  .  .  .  .
      0 |  X  X  .  .  .  .               0 |  X  .  .  .  .  X
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,2,4,5,8,10,13,17,20,26}  dist^2  {1,2,4,5,8,10,13,20,25,26}
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  .  .  .  X  .               2 |  .  .  X  .  X  .
      1 |  X  .  .  .  .  X               1 |  X  .  .  .  .  X
      0 |  X  .  X  .  .  .               0 |  X  .  .  .  .  .
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,2,4,5,8,10,17,20,25,26}  dist^2  {1,2,4,5,8,10,17,20,25,26}
           0  1  2  3  4  5                    0  1  2  3  4  5
         -------------------                 -------------------
      2 |  .  .  X  .  .  X               2 |  X  .  .  .  .  X
      1 |  .  .  .  .  .  .               1 |  .  .  .  .  .  .
      0 |  X  X  .  .  .  X               0 |  X  .  .  X  X  .
      y /-------------------              y /-------------------
        x  0  1  2  3  4  5                 x  0  1  2  3  4  5
    {1,4,5,8,9,13,16,20,25,29}  dist^2  {1,4,5,8,9,13,16,20,25,29}
  .

Hugo Pfoertner, Table of n, a(n) for n = 1..91, rows 1..13 of triangle, flattened

Cf. A193838, A193839, A271490, A325677, A335232, A351700, A351701.

Completed row 8 and new rows 9-12 from Hugo Pfoertner, Jul 12 2022

A353447 a(n) is the number of tetrapods standing on the four edges of an n X n grid, so that no two feet are the same distance apart and no foot is on a corner. Tetrapods with congruent footprints are counted only once.

Original entry on oeis.org

0, 0, 1, 11, 40, 105, 190, 379, 616, 987, 1426, 2139, 2964, 4130, 5403, 7180, 9155, 11716, 14458, 18092, 22037, 26808, 31793, 38343, 45060, 53184, 61613, 71878, 82466, 95368, 108195, 123790, 140040, 158457, 177405, 200020, 223039, 248769, 275214, 306411, 337645

Offset: 3

Rainer Rosenthal, Apr 20 2022

nonn

If we name the tetrapod's footprints "mini-frame", we can say that mini-frames span their grid, i.e., there is no smaller grid for them. Every corner-less set of points with distinct distances in a smallest possible n X n grid contains at least one mini-frame.

			  .
     . C .           a(3) = 0              . . . C .
     D . B   <===  since AB = CD           . . . . .
     . A .         is forbidden            . . . . B
                                           . . . . .
                        . C . .            D . . . .
      a(4) = 0  ===>    ? . . .            . A . . .
    (there is no        ? . . B         ______________
     space for D)       . A . .            a(5) = 1
                                     (No other solutions)
  .
    . . . . .           The tetrapod has 6 distinct
    D . . . .           squared distances 4, 5, 10,
    . . . . C   <=====  13, 17, 18, but it uses only
    . . . . .           three edges of the 5 X 5 grid.
    . A . B .           (Not allowed.)
  .

Hugo Pfoertner, Table of n, a(n) for n = 3..200

Cf. A193838, A271490, A335232, A351699, A351700, A353532.

The general case without excluding the corners of the grid rectangle is covered in A354700 and A354701.

a(23) and beyond from Hugo Pfoertner, Apr 20 2022

cp's OEIS Frontend

A271490 Size of maximal subset of points of n X n grid such that no two points are at the same distance.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A351699 T(n,k) is the number of non-congruent maximal subsets of a grid of n X k lattice points (k <= n), such that no two points are at the same distance, and the points do not fit into a smaller grid. The size of the subsets is given by A351700. T(n,k) and A351700 are triangles read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A353447 a(n) is the number of tetrapods standing on the four edges of an n X n grid, so that no two feet are the same distance apart and no foot is on a corner. Tetrapods with congruent footprints are counted only once.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions