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A066062
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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.
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29
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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
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OFFSET
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0,3
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COMMENTS
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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.
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)
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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, [])
(C) See Martin Fuller link in A158449
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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