%I #70 Sep 04 2024 12:58:08
%S 1,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,
%T 10,10,10,10,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,
%U 13,13,13,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,16
%N Size of smallest subset S of N={0,1,2,...,n} such that S-S=N, where S-S={abs(i-j) | i,j in S}.
%C It is easy to show that a(n+1) must be no larger than a(n)+1. Problem: Can a(n+1) ever be smaller than a(n)?
%C Problem above solved in A103300. a(137) smaller than a(136).
%C Except for initial term, round(sqrt(3*n + 9/4)) up to n=51. See A308766 for divergences up to n=213. See A326499 for a list of best known solutions.
%C From Ed Pegg Jr, Jun 23 2019: (Start)
%C Minimal marks for a sparse ruler of length n.
%C Minimal vertices in a graceful graph with n edges. (End)
%H Ed Pegg Jr, <a href="/A046693/b046693.txt">Table of n, a(n) for n = 0..213</a>
%H Andrew Granville and Friedrich Roesler, <a href="https://dms.umontreal.ca/~andrew/PDF/Roesler.pdf">The set of differences of a given set</a>
%H Andrew Granville and Friedrich Roesler, <a href="https://www.jstor.org/stable/2589556">The set of differences of a given set</a>, Amer. Math. Monthly, 106 (1999), 338-344.
%H J. Leech, <a href="https://doi.org/10.1112/jlms/s1-31.2.160">On the representation of 1, 2, ..., n by differences</a>, J. Lond. Math. Soc. 31 (1956), 160-169.
%H Ed Pegg Jr, <a href="/A046693/a046693_1.jpg">Best known values A046693(n)-round(sqrt(3*n+9/4)) up to 1750<</a>
%H Aleksi Saarela and Aleksi Vanhatalo, <a href="https://arxiv.org/abs/2408.16335">A Connection Between Unbordered Partial Words and Sparse Rulers</a>, arXiv:2408.16335 [math.CO], 2024. See p. 17.
%e a(10)=6 since all integers in {0,1,2...10} are differences of elements of {0,1,2,3,6,10}, but not of any 5-element set.
%e a(17)=7 since all integers in {0,1,2...17} are differences of elements of {0,1,8,11,13,15,17}, but not of any 6-element set.
%e In other words, {0,1,8,11,13,15,17} is a restricted difference basis w.r.t. A004137(7)=17.
%t Prepend[Table[Round[Sqrt[3*n+9/4]]+If[MemberQ[A308766,n],1,0],{n,1,213}],1]
%Y Cf. A004137, A103300, A308766, A309407, A326499. - _Ed Pegg Jr_, Sep 14 2019
%K nonn,hard
%O 0,2
%A _John W. Layman_