

A066062


Number of distinct subsets S of T={0,1,2,...,n} such that each element of T is the sum of two elements of S.


13



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
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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 2bases 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

Table of n, a(n) for n=0..26.
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 ErdosTurán conjecture, J. Number Theory 102 (2003), no. 2, 339352.
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) 650667.


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}] (* JeanFrançois Alcover, Nov 05 2018 *)


CROSSREFS

Cf. A008929, A066063.
Cf. A158291.  Steven Finch, Mar 15 2009
Sequence in context: A329702 A222855 A171682 * A008929 A164047 A158291
Adjacent sequences: A066059 A066060 A066061 * A066063 A066064 A066065


KEYWORD

nonn,more


AUTHOR

John W. Layman, Dec 01 2001


STATUS

approved



