A045691 - OEIS

%I #17 Jan 29 2022 12:48:53

%S 0,1,1,3,5,11,19,41,77,159,307,625,1231,2481,4921,9883,19689,39455,

%T 78751,157661,315015,630337,1260049,2520723,5040215,10081661,20160841,

%U 40324163,80643405,161291731,322573579,645157041,1290294393,2580608475,5161177495

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

%C From _Gus Wiseman_, Jan 22 2022: (Start)

%C 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:

%C   {1}  {2}  {3}    {4}      {5}        {6}

%C             {1,3}  {1,4}    {1,5}      {1,6}

%C             {2,3}  {3,4}    {2,5}      {2,6}

%C                    {1,3,4}  {3,5}      {4,6}

%C                    {2,3,4}  {4,5}      {5,6}

%C                             {1,3,5}    {1,4,6}

%C                             {1,4,5}    {1,5,6}

%C                             {2,3,5}    {2,5,6}

%C                             {3,4,5}    {3,4,6}

%C                             {1,3,4,5}  {3,5,6}

%C                             {2,3,4,5}  {4,5,6}

%C                                        {1,3,4,6}

%C                                        {1,3,5,6}

%C                                        {1,4,5,6}

%C                                        {2,3,4,6}

%C                                        {2,3,5,6}

%C                                        {3,4,5,6}

%C                                        {1,3,4,5,6}

%C                                        {2,3,4,5,6}

%C (End)

%F 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.

%t 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 *)

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

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

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

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

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

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

%Y - Strict partitions of this type are counted by A350840.

%Y - For differences instead of quotients we have A350842, strict A350844.

%Y - Partitions not of this type are counted by A350846, ranked by A350845.

%Y A000740 = relatively prime subsets of {1..n} containing n.

%Y A002843 = compositions with all adjacent quotients >= 1/2.

%Y A050291 = double-free subsets of {1..n}.

%Y A154402 = partitions with all adjacent quotients 2.

%Y A308546 = double-closed subsets of {1..n}, with maximum: shifted right.

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

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

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

%K nonn

%O 0,4

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

%E More terms from _Sean A. Irvine_, Mar 18 2021