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.

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

Author

Hugo Pfoertner, Apr 08 2022

Keywords

• nonn
• tabl
• hard

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A193838, A271490, A351699, A351700.

Extensions

a(55)=T(10,10) corrected by Hugo Pfoertner, Nov 06 2022