
EXAMPLE

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
