

A045691


Number of binary words of length n with autocorrelation function 2^(n1)+1.


8



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

0,4


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)


LINKS

Table of n, a(n) for n=0..34.


FORMULA

a(2*n1) = 2*a(2*n2)  a(n) for n >= 2; a(2*n) = 2*a(2*n1) + a(n) for n >= 2.


MATHEMATICA

Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&And@@Table[#[[i1]]/#[[i]]!=1/2, {i, 2, Length[#]}]&]], {n, 0, 15}] (* Gus Wiseman, Jan 22 2022 *)


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 = doublefree subsets of {1..n}.
A154402 = partitions with all adjacent quotients 2.
A308546 = doubleclosed subsets of {1..n}, with maximum: shifted right.
A323092 = doublefree integer partitions, ranked by A320340, strict A120641.
A326115 = maximal doublefree subsets of {1..n}.
Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.
Sequence in context: A089098 A129384 A131887 * A045961 A293820 A281380
Adjacent sequences: A045688 A045689 A045690 * A045692 A045693 A045694


KEYWORD

nonn


AUTHOR

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


EXTENSIONS

More terms from Sean A. Irvine, Mar 18 2021


STATUS

approved



