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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. 7
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..13.

P. Erdős and R. K. Guy, Distinct distances between lattice points, Elemente der Mathematik 25 (1970), 121-123.

Sean A. Irvine, Java program (github)

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).

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

CROSSREFS

Cf. A193839, A003022.

See A271490 for the inverse function.

Sequence in context: A062462 A033948 A285514 * A117730 A174328 A272570

Adjacent sequences:  A193835 A193836 A193837 * A193839 A193840 A193841

KEYWORD

nonn,hard,more

AUTHOR

Hugo Pfoertner, Aug 06 2011

EXTENSIONS

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

STATUS

approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)