OFFSET
0,3
COMMENTS
This sequence may be equivalent to A008929, but has a somewhat different definition. The size of the smallest subset counted by this sequence, for a given n, is given in A066063.
From Steven Finch, Mar 15 2009: (Start)
Such sets S are called additive 2-bases for {0,1,2,...,n}.
a(n) is also the number of symmetric numerical sets S with atom monoid A(S) equal to {0, 2n+2, 2n+3, 2n+4, 2n+5, ...}. (End)
LINKS
Martin Fuller, Table of n, a(n) for n = 0..65
Steven R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
Georges Grekos, Labib Haddad, Charles Helou, and Jukka Pihko, On the Erdos-Turán conjecture, J. Number Theory 102(2) (2003), 339-352.
Jeremy Marzuola and Andy Miller, Counting numerical sets with no small atoms, arXiv:0805.3493 [math.CO], 2008. - Steven Finch, Mar 15 2009
Jeremy Marzuola and Andy Miller, Counting numerical sets with no small atoms, J. Combin. Theory A 117 (6) (2010) 650-667.
Melvyn B. Nathanson, Problems in additive number theory, VII: The structure of additive h-bases for n, arXiv:2605.26425 [math.NT], 2026.
FORMULA
a(n) = 2*a(n-1) - A158449(n) [adapted from A164047]. - Martin Fuller, Sep 13 2023 [corrected by Martin Fuller, Apr 21 2026]
a(n) >= A001405(n). - Michael Chu, Oct 15 2023
EXAMPLE
For n=2, the definition obviously requires that S contain both 0 and 1. The only subsets of {0,1,2} that do this are {0,1} and {0,1,2}. For both of these, we have 0=0+0, 1=0+1, 2=1+1, so a(2)=2.
MATHEMATICA
a[n_] := Module[{},
T = Range[0, n];
ST = Subsets[T, {Floor[n^(2/3)], n+1}];
selQ[S_] := Intersection[T, Total /@ Tuples[S, {2}]] == T;
SST = Select[ST, selQ]; min = Min[Length /@ SST];
SST // Length
];
Table[an = a[n]; Print["a(", n, ") = ", an, " min = ", min]; an, {n, 0, 24}] (* Jean-François Alcover, Nov 05 2018 *)
PROG
(Python)
def sums(s): return set((si+sj) for i, si in enumerate(s) for sj in s[i:])
def b(i, n, s):
if sums(s) >= set(range(n+1)): return 2**(n+1-i)
if i > n: return 0
return b(i+1, n, s) + b(i+1, n, s+[i])
def a(n): return b(0, n, [])
print([a(n) for n in range(15)]) # Michael S. Branicky, Jan 15 2022
(C) // See Martin Fuller link in A158449
CROSSREFS
KEYWORD
nonn
AUTHOR
John W. Layman, Dec 01 2001
EXTENSIONS
a(27)-a(30) from Michael S. Branicky, Jan 15 2022
a(31) onwards from Martin Fuller, Sep 13 2023
STATUS
approved