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

%I A193838 #83 Jul 17 2022 00:56:17
%S A193838 1,2,3,4,5,6,7,9,10,11,13,15,16,18
%N A193838 Size k of smallest square of k X k lattice points from which n points with distinct mutual distances can be chosen.
%C A193838 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
%C A193838 Upper bounds for a(15) to a(18): 20, 22, 24, 27. - _Fausto A. C. Cariboni_, Jul 16 2022
%D A193838 R. K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer New York, 2004, F2, 367-368.
%D A193838 Keith F. Lynch, Posting to Math Fun Mailing List, Apr 02 2016.
%H A193838 P. Erdős and R. K. Guy, <a href="http://dx.doi.org/10.5169/seals-27359">Distinct distances between lattice points</a>, Elemente der Mathematik 25 (1970), 121-123.
%H A193838 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a193/A193838.java">Java program</a> (github)
%H A193838 Dmitry Kamenetsky, <a href="/A193838/a193838_1.txt">Best known solutions for n <= 13</a>.
%H A193838 Dmitry Kamenetsky, <a href="https://puzzling.stackexchange.com/questions/109624/7x7-golomb-square">7x7 Golomb Square</a>, Puzzling StackExchange, April 2021.
%H A193838 Matt Parker, <a href="http://www.think-maths.co.uk/uniquedistance">Unique Distancing Puzzle</a>.
%H A193838 Samuel B. Reid, <a href="/A193838/a193838.png">The unique solution that causes a(7) to be 7</a>.
%H A193838 Wolfram Demonstration Project, <a href="http://demonstrations.wolfram.com/NoRepeatedDistances/">No Repeated Distances.</a>
%H A193838 A. Zimmermann, <a href="http://azspcs.com/Contest/PointPacking/FinalReport">Al Zimmermann's Programming Contests: Point Packing.</a> (Oct 10, 2009).
%e A193838 a(1) is the degenerate case of a single point, a(2)=2 is trivial.
%e A193838 a(3)=3: The points ((1,2),(3,1),(3,2)) have distinct mutual squared distances 1, 4, 5.
%e A193838 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.
%e A193838 a(10)=11, only two subsets barring symmetry:
%e A193838   {(0,0), (0,2), (0,3), (0,7), (1,10), (5,4), (6,0), (8,7), (9,8), (10, 10)},
%e A193838   {(0,0), (0,6), (0,7), (1,2), (4,10), (7,8), (7,10), (9,2), (9,6), (10,5)}.
%e A193838 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)}
%e A193838 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
%e A193838 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
%Y A193838 Cf. A193839, A003022.
%Y A193838 See A271490 for the inverse function.
%K A193838 nonn,hard,more
%O A193838 1,2
%A A193838 _Hugo Pfoertner_, Aug 06 2011
%E A193838 a(10)-a(11) corrected by _Ehit Dinesh Agarwal_, May 28 2020
%E A193838 a(12) from _Sean A. Irvine_, Jul 13 2020
%E A193838 a(13) from _Bert Dobbelaere_, Sep 20 2020
%E A193838 a(14) from _Fausto A. C. Cariboni_, Jul 16 2022