OFFSET
0,2
COMMENTS
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)?
Problem above solved in A103300. a(137) smaller than a(136).
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.
From Ed Pegg Jr, Jun 23 2019: (Start)
Minimal marks for a sparse ruler of length n.
Minimal vertices in a graceful graph with n edges. (End)
LINKS
Ed Pegg Jr, Table of n, a(n) for n = 0..213
Andrew Granville and Friedrich Roesler, The set of differences of a given set
Andrew Granville and Friedrich Roesler, The set of differences of a given set, Amer. Math. Monthly, 106 (1999), 338-344.
J. Leech, On the representation of 1, 2, ..., n by differences, J. Lond. Math. Soc. 31 (1956), 160-169.
Aleksi Saarela and Aleksi Vanhatalo, A Connection Between Unbordered Partial Words and Sparse Rulers, arXiv:2408.16335 [math.CO], 2024. See p. 17.
EXAMPLE
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.
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.
In other words, {0,1,8,11,13,15,17} is a restricted difference basis w.r.t. A004137(7)=17.
MATHEMATICA
Prepend[Table[Round[Sqrt[3*n+9/4]]+If[MemberQ[A308766, n], 1, 0], {n, 1, 213}], 1]
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
STATUS
approved