A066062 Number of subsets S of T={0,1,2,...,n} such that each element of T is the sum of two (not necessarily distinct) elements of S.

1, 1, 2, 3, 6, 10, 20, 37, 73, 139, 275, 533, 1059, 2075, 4126, 8134, 16194, 32058, 63910, 126932, 253252, 503933, 1006056, 2004838, 4004124, 7987149, 15957964, 31854676, 63660327, 127141415, 254136782, 507750109, 1015059238, 2028564292, 4055812657, 8107052520

Offset: 0

Author

John W. Layman, Dec 01 2001

Keywords

• nonn

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)

Examples

			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.
		

Links

• Martin Fuller, Table of n, a(n) for n = 0..65
• S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
• G. Grekos, L. Haddad, C. Helou, and J. Pihko, On the Erdos-Turán conjecture, J. Number Theory 102 (2003), no. 2, 339-352.
• J. Marzuola and A. Miller, Counting numerical sets with no small atoms, arXiv:0805.3493 [math.CO], 2008. - _Steven Finch_, Mar 15 2009
• J. Marzuola and A. Miller, Counting numerical sets with no small atoms, J. Combin. Theory A 117 (6) (2010) 650-667.

Crossrefs

Cf. A008929, A066063, A158449, A164047.

Cf. A158291. - Steven Finch, Mar 15 2009

Programs

C

See Martin Fuller link in A158449

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 *)

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

Formula

a(n) = 2*a(n-1) - A158449(n) [adapted from A164097]. - Martin Fuller, Sep 13 2023

a(n) >= A001405(n). - Michael Chu, Oct 15 2023

Extensions

a(27)-a(30) from Michael S. Branicky, Jan 15 2022

a(31) onwards from Martin Fuller, Sep 13 2023