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A045691
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Number of binary words of length n with autocorrelation function 2^(n-1)+1.
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8
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0, 1, 1, 3, 5, 11, 19, 41, 77, 159, 307, 625, 1231, 2481, 4921, 9883, 19689, 39455, 78751, 157661, 315015, 630337, 1260049, 2520723, 5040215, 10081661, 20160841, 40324163, 80643405, 161291731, 322573579, 645157041, 1290294393, 2580608475, 5161177495
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OFFSET
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0,4
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COMMENTS
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Also the number of subsets of {1..n} containing n but without adjacent elements of quotient 1/2. The Heinz numbers of these sets are a subset of the squarefree terms of A320340. For example, the a(1) = 1 through a(6) = 19 subsets are:
{1} {2} {3} {4} {5} {6}
{1,3} {1,4} {1,5} {1,6}
{2,3} {3,4} {2,5} {2,6}
{1,3,4} {3,5} {4,6}
{2,3,4} {4,5} {5,6}
{1,3,5} {1,4,6}
{1,4,5} {1,5,6}
{2,3,5} {2,5,6}
{3,4,5} {3,4,6}
{1,3,4,5} {3,5,6}
{2,3,4,5} {4,5,6}
{1,3,4,6}
{1,3,5,6}
{1,4,5,6}
{2,3,4,6}
{2,3,5,6}
{3,4,5,6}
{1,3,4,5,6}
{2,3,4,5,6}
(End)
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LINKS
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FORMULA
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a(2*n-1) = 2*a(2*n-2) - a(n) for n >= 2; a(2*n) = 2*a(2*n-1) + a(n) for n >= 2.
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MATHEMATICA
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Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&And@@Table[#[[i-1]]/#[[i]]!=1/2, {i, 2, Length[#]}]&]], {n, 0, 15}] (* Gus Wiseman, Jan 22 2022 *)
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CROSSREFS
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If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:
- The version with quotients >= 1/2 is A045690(n+1), partitions A342094.
- Strict partitions of this type are counted by A350840.
- For differences instead of quotients we have A350842, strict A350844.
- Partitions not of this type are counted by A350846, ranked by A350845.
A000740 = relatively prime subsets of {1..n} containing n.
A002843 = compositions with all adjacent quotients >= 1/2.
A050291 = double-free subsets of {1..n}.
A154402 = partitions with all adjacent quotients 2.
A308546 = double-closed subsets of {1..n}, with maximum: shifted right.
A326115 = maximal double-free subsets of {1..n}.
Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.
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KEYWORD
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nonn
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AUTHOR
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Torsten Sillke (torsten.sillke(AT)lhsystems.com)
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EXTENSIONS
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STATUS
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approved
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