A351700 -id:A351700 - cp's OEIS frontend

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.

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

A351701 Smallest maximum of the distinct squared distances between any two of the points taken over all possible solutions, written as triangle T(n,k) with problem size and number of points given by the corresponding A351700.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 10, 13, 16, 16, 16, 20, 20, 25, 25, 26, 25, 25, 41, 36, 36, 36, 40, 37, 37, 72, 49, 49, 49, 49, 58, 50, 50, 52, 64, 64, 64, 64, 64, 64, 73, 80, 74, 81, 81, 81, 81, 81, 90, 82, 82, 100, 113, 100, 100, 100, 100, 109, 100, 100, 109, 106, 104, 149

Offset: 1

Hugo Pfoertner, Apr 08 2022

This sequence considers only solutions that do not fit into a smaller grid, as in A351699. - Fausto A. C. Cariboni, Nov 08 2022

			Correspondence between the triangle of A351700 and T(n,k), with terms of this sequence shown delimited by parenthesis.
   n\k 1   2   3   4   5   6   7   8   9  10  11
   1:  1   |   |   |   |   |   |   |   |   |   |
     ( 0)  |   |   |   |   |   |   |   |   |   |
   2:  2   2   |   |   |   |   |   |   |   |   |
     ( 1   2)  |   |   |   |   |   |   |   |   |
   3:  2   3   3   |   |   |   |   |   |   |   |
     ( 4   5   8)  |   |   |   |   |   |   |   |
   4:  3   4   4   4   |   |   |   |   |   |   |
     ( 9  10  10  13)  |   |   |   |   |   |   |
   5:  3   4   4   5   5   |   |   |   |   |   |
     (16  16  16  20  20)  |   |   |   |   |   |
   6:  3   4   5   5   5   6   |   |   |   |   |
     (25  25  26  25  25  41)  |   |   |   |   |
   7:  4   5   5   6   6   6   7   |   |   |   |
     (36  36  36  40  37  37  72)  |   |   |   |
   8:  4   5   5   6   7   7   7   7   |   |   |
     (49  49  49  49  58  50  50  52)  |   |   |
   9:  4   5   6   6   7   7   8   8   8   |   |
     (64  64  64  64  64  64  73  80  74)  |   |
  10:  4   6   6   7   7   8   8   8   9   9   |
     (81  81  81  81  81  90  82  82 100 113)  |
  11:  4   6   6   7   8   8   8   9   9   9  10
    (100 100 100 100 109 100 100 109 106 104 149)
.
T(6,6) = a(21) = 41:
There are only 2 essentially different point configurations of A351700(21) = 6 selected grid points:
[(0,0), (1,0), (2,5), (3,1), (5,3), (5,5)] with the corresponding list of squared distances {1, 4, 5, 8, 9, 10, 13, 17, 20, 25, 26, 29, 34, 41, 50},
and [(0,0),( 0, 3),( 0, 5),( 3, 2),( 4, 1),( 5, 1)] with squared distances
{1, 2, 4, 5, 9, 10, 13, 17, 18, 20, 25, 26, 29, 32, 41}.
The maximum of squared distances in the second configuration between the points (0,5) and (5,1) is 41, whereas the squared distance in the first configuration is 50, made by the corner points (0,0) and (5,5). Thus a(21) = min(41,50) = 41.
.
T(11,11) = a(66) = 149. The two possible configurations with 10 points on the quadratic grid with 11 X 11 points are given in the comments of A193838 or A271490. The first configuration uses the two corner points (0,0) and (10,10) with squared distance 200, whereas in the other configuration a squared distance of 149 between the points (0,0) and (7,10) is maximal. Thus a(66) = min(200,149) = 149.

Fausto A. C. Cariboni, Rows n = 1..16, flattened

Cf. A193838, A271490, A351699, A351700.

a(55)=T(10,10) corrected by Hugo Pfoertner, Nov 06 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

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.

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A351701 Smallest maximum of the distinct squared distances between any two of the points taken over all possible solutions, written as triangle T(n,k) with problem size and number of points given by the corresponding A351700.

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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.

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