A271490 -id:A271490 - cp's OEIS frontend

A193838 Size k of smallest square of k X k lattice points from which n points with distinct mutual distances can be chosen.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 16, 18

Offset: 1

Hugo Pfoertner, Aug 06 2011

Upper bounds for a(14) to a(26): 18, 21, 24, 26, 28, 29, 33, 36, 37, 40, 43, 46, 49. These have been obtained from the results of the Al Zimmermann competition. - Dmitry Kamenetsky, Apr 23 2021

Upper bounds for a(15) to a(18): 20, 22, 24, 27. - Fausto A. C. Cariboni, Jul 16 2022

			a(1) is the degenerate case of a single point, a(2)=2 is trivial.
a(3)=3: The points ((1,2),(3,1),(3,2)) have distinct mutual squared distances 1, 4, 5.
a(8)=9 is the first square for which k>n: ((1,1), (1,4), (2,2), (6,1), (7,6), (7,7), (9,2), (9,4)) have 7*8/2=28 mutual squared distances: 1, 2, 4, 5, 8, 9, 10, 13, 17, 18, 20, 25, 26, 29, 34, 37, 40, 41, 45, 49, 50, 53, 61, 64, 65, 68, 72, 73, and no configuration of 8 points fitting on an 8 X 8 square exists.
a(10)=11, only two subsets barring symmetry:
  {(0,0), (0,2), (0,3), (0,7), (1,10), (5,4), (6,0), (8,7), (9,8), (10, 10)},
  {(0,0), (0,6), (0,7), (1,2), (4,10), (7,8), (7,10), (9,2), (9,6), (10,5)}.
a(11)=13, one of the four subsets of the 12 X 13 grid, barring symmetry: {(0,0), (0,1), (0,9), (0,12), (2,0), (5,3), (6,12), (7,0), (8,4), (10,10), (11,11)}
a(12)=15 is satisfied by {(0,0), (1,0), (1,12), (3,0), (7,0), (7,14), (9,4), (12,11), (13,3), (13,8), (14,2), (14,13)}. - _Sean A. Irvine_, Jul 13 2020
a(13)=16 is satisfied by {(1,1), (2,2), (2,16), (4,14), (6,14), (7,16), (8,8), (11,2), (11,5), (13,15), (13,16), (16,1), (16,8)}. - _Bert Dobbelaere_, Sep 20 2020

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.
Sean A. Irvine, Java program (github)
Dmitry Kamenetsky, Best known solutions for n <= 13.
Dmitry Kamenetsky, 7x7 Golomb Square, Puzzling StackExchange, April 2021.
Matt Parker, Unique Distancing Puzzle.
Samuel B. Reid, The unique solution that causes a(7) to be 7.
Wolfram Demonstration Project, No Repeated Distances.
A. Zimmermann, Al Zimmermann's Programming Contests: Point Packing. (Oct 10, 2009).

Cf. A193839, A003022.

See A271490 for the inverse function.

a(10)-a(11) corrected by Ehit Dinesh Agarwal, May 28 2020
a(12) from Sean A. Irvine, Jul 13 2020
a(13) from Bert Dobbelaere, Sep 20 2020
a(14) from Fausto A. C. Cariboni, Jul 16 2022

A303331 a(n) is the minimum size of a square integer grid allowing all triples of n points to form triangles of different areas.

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 9, 11, 15, 19

Offset: 1

Bert Dobbelaere, Apr 30 2018

Place n points on integer coordinates of a square grid of dimension L x L, so that no 3 points are collinear and the areas of the triangles formed by the binomial(n,3) possible triples are all different. a(n) is the minimum L for which this can be achieved. (Note that there are (L+1)^2 lattice points.)
The fact that all triangle areas are multiples of 1/2 and the maximum area supported by the grid is L^2/2 provides us with a lower bound for a(n).
The problem can be considered a 2-dimensional extension of a Golomb ruler and scales to higher dimensions. In general, consider k points on integer coordinates of a D-dimensional hypercube, forming binomial(k,D+1) D-simplices of which the volumes are all different. For D = 1, we recognize the Golomb ruler; for D = 2, we have the problem defined above.
Just like for Costas arrays (another 2-D extension of Golomb rulers), no 2 displacement vectors can be identical, as the diagonal of a parallelogram cuts the shape in triangles of identical area.
a(11) <= 24, a(12) <= 29. - Hugo Pfoertner, Nov 05 2018

			For n = 5, a solution satisfying unequal triangle areas is {(0,4),(1,1),(3,0),(3,3),(4,3)}, which can be verified by considering the binomial(5,3) = 10 possible triangles by selecting vertices from this set. Each coordinate is contained in the range [0..4]. No smaller solution is possible without creating areas that are no longer unique, hence a(5) = 4.
From _Jon E. Schoenfield_, Apr 30 2018: (Start)
Illustration of the above solution:
                        vertices   area
  4   1  .  .  .  .     --------   ----
                          1 2 3     2.5
  3   .  .  .  4  5       1 2 4     4.0
                          1 2 5     5.5
  2   .  .  .  .  .       1 3 4     4.5
                          1 3 5     6.5
  1   .  2  .  .  .       1 4 5     0.5
                          2 3 4     3.0
  0   .  .  .  3  .       2 3 5     3.5
  y                       2 4 5     1.0
    x 0  1  2  3  4       3 4 5     1.5
(End)

IBM Research, Ponder This Challenge October 2018. Similar problem for rectangular grids.

For optimal Golomb rulers, see A003022.

Cf. A193838, A193839, A271490, A322740.

def addNewAreas(plist, used_areas):
    # Attempt to add last point in plist. 'used_areas' contains all (areas*2)
    # between preplaced points and is updated
    m=len(plist)
    for i in range(m-2):
        for j in range(i+1,m-1):
            k=m-1
            p,q,r=plist[i],plist[j],plist[k]
            # Area = s/2 - using s enables us to use integers
            s=(p[0]*q[1]-p[1]*q[0]) + (q[0]*r[1]-q[1]*r[0]) + (r[0]*p[1]-r[1]*p[0])
            if s<0:
                s=-s
            if s in used_areas:
                return False # Area not unique
            else:
                used_areas[s]=True
    return True
def solveRecursively(n, L, plist, used_areas):
    m=len(plist)
    if m==n:
        #print plist (uncomment to show solution)
        return True
    newlist=plist+[None]
    if m>0:
        startidx=(L+1)*plist[m-1][0] + plist[m-1][1] + 1
    else:
        startidx=0
    for idx in range(startidx, (L+1)**2):
            newlist[m]=( idx/(L+1) , idx%(L+1) )
            newareas=dict(used_areas)
            if addNewAreas(newlist, newareas):
                if solveRecursively(n, L, newlist, newareas):
                    return True
    return False
def A303331(n):
    L=0
    while not solveRecursively(n, L, [], {0:True}):
        L+=1
    return L
def A303331_list(N):
    return [A303331(n) for n in range(1,N+1)]
# Bert Dobbelaere, May 01 2018

a(n+1) >= a(n) (trivial).

a(n) >= sqrt(n*(n-1)*(n-2)/6) for n >= 2 (proven lower bound).

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

A335232 Number of largest subsets of the set of points in an n X n square grid, such that no two points are at the same distance.

Original entry on oeis.org

1, 6, 40, 184, 280, 16, 8, 26800, 4376, 416, 16, 27488, 536, 587640, 10192, 128

Offset: 1

Ehit Dinesh Agarwal, May 27 2020

Ehit Dinesh Agarwal, C program
P. Erdős and R. K. Guy, Distinct distances between lattice points, Elemente der Mathematik 25 (1970), 121-123.
Matt Parker, MPMP: Unique Distancing Problem, Youtube video, May 2020.
Giovanni Resta, The essentially unique 7 x 7 configuration

Cf. A193838, A193839.
Size of largest subset in A271490.
Cf. A351699 (generalization to grid rectangles, counting congruent configurations only once).

a(14)-a(16) from Fausto A. C. Cariboni, Jul 03 2022

A351700 T(n,k) is the maximum number of points that can be chosen from a rectangle of n X k lattice points such that their mutual distances are distinct, where T(n,k) is a triangle read by rows, 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Mar 05 2022

			The triangle begins:
  1
  2  2
  2  3  3
  3  4  4  4
  3  4  4  5  5
  3  4  5  5  5  6
  4  5  5  6  6  6  7
  4  5  5  6  7  7  7  7
  4  5  6  6  7  7  8  8  8
  4  6  6  7  7  8  8  8  9  9
  4  6  6  7  8  8  8  9  9  9 10
  5  6  7  7  8  9  9  9  9 10 10 10
  5  6  7  8  8  9  9 10 10 10 10 11 11

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

First occurrence of n in first column: A227590.
Main diagonal: A271490.
Cf. A003022, A193838, A351699.

T(13,2)=a(80) and T(13,8)=a(86) corrected by Fausto A. C. Cariboni, Jul 10 2022

A322740 Least area of a grid rectangle from which n >= 3 grid points can be chosen such that the binomial(n,3) triangles formed by any 3 of the points have distinct areas > 0.

Original entry on oeis.org

1, 4, 15, 30, 65, 120, 198, 342

Offset: 3

Hugo Pfoertner, Dec 24 2018

Finding configurations of 11 points that can placed on a rectangle with area a(11) <= 600 had been the topic of the October 2018 Ponder This Challenge, based on a suggestion by Bert Dobbelaere.

The solution with the least area found by Hermann Jurksch gives an upper bound of a(11) <= 528. a(12) <= 841 from private communication with H. Jurksch.

			a(3) = 1 (trivial).
a(4) = 4: Choosing (0,0),(0,2),(1,0),(2,1) gives 4 triangles with areas (1/2)*{1 2 3 4}.
a(5) = 15: Choosing (0,0),(0,1),(1,5),(2,0),(3,2) from a 3 X 5 rectangle gives 10 triangles with areas (1/2)*{1 2 3 4 5 7 9 10 11 13}.
a(6) = 30: (0,5),(1,0),(2,6),(3,0),(5,3),(5,4) is a solution on the 5 X 6 rectangle.
a(7) = 65: (0,3),(1,5),(7,0),(11,1),(11,5),(12,4),(13,2) is a solution on the 13 X 5 rectangle.
a(8) = 120: There are two minimal solutions, (0,12),(1,0),(2,15),(3,0),(6,1),(6,4),(7,7),(8,14) on the 8 X 15 rectangle giving 56 triangles with areas (1/2)*{1 2 3 4 5 6 8 9 11 12 14 15 17 18 20 21 22 23 24 26 27 28 29 30 31 33 34 35 37 38 39 40 42 43 46 47 48 49 54 61 62 63 64 67 69 71 74 76 79 80 83 89 91 98 100 102}, and (0,7),(0,9),(1,1),(5,1),(6,10),(8,0),(9,12),(10,3) on the 10 X 12 rectangle with triangle areas {2 3 4 8 9 10 11 12 13 16 17 18 19 20 21 23 24 25 26 28 29 32 34 36 37 38 39 40 41 42 43 44 46 47 49 50 51 53 54 55 56 59 62 66 68 71 74 75 79 83 84 85 86 87 103 105}.
a(9) = 198: The unique solution, up to rotation and reflection, is (0,3),(1,9),(2,0),(3,10),(4,11),(12,2),(17,11),(18,1),(18,4) on an 18 X 11 rectangle. The list of the 84 corresponding triangle areas is given in A322741.
a(10) = 342: The unique solution, up to rotation and reflection, is (0,3),(1,9),(2,18),(5,0),(5,10),(12,17),(15,13),(17,4),(18,0),(19,5) on a 19 X 18 rectangle. The list of the 120 corresponding triangle areas is given in A322742.

IBM Research, Ponder This Challenge - October 2018.
IBM Research, Triangle Game.

Cf. A193838, A271490, A303331, A322741, A322742.

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

cp's OEIS Frontend

A193838 Size k of smallest square of k X k lattice points from which n points with distinct mutual distances can be chosen.

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A303331 a(n) is the minimum size of a square integer grid allowing all triples of n points to form triangles of different areas.

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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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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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A335232 Number of largest subsets of the set of points in an n X n square grid, such that no two points are at the same distance.

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A351700 T(n,k) is the maximum number of points that can be chosen from a rectangle of n X k lattice points such that their mutual distances are distinct, where T(n,k) is a triangle read by rows, 1 <= k <= n.

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A322740 Least area of a grid rectangle from which n >= 3 grid points can be chosen such that the binomial(n,3) triangles formed by any 3 of the points have distinct areas > 0.

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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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