A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

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

Offset: 0

Author

Torsten Sillke (torsten.sillke(AT)lhsystems.com)

Keywords

• nonn

Comments

From Gus Wiseman, Jan 22 2022: (Start)

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)

Crossrefs

If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:

- The version with quotients <= 1/2 is A018819, partitions A000929.

- The version with quotients < 1/2 is A040039, partitions A342098.

- The version with quotients >= 1/2 is A045690(n+1), partitions A342094.

- The version with quotients > 1/2 is A045690, partitions A342096.

- Partitions of this type are counted by A350837, ranked by A350838.

- 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.

A323092 = double-free integer partitions, ranked by A320340, strict A120641.

A326115 = maximal double-free subsets of {1..n}.

Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.

Programs

Mathematica

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 *)

Formula

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.

Extensions

More terms from Sean A. Irvine, Mar 18 2021