cp's OEIS Frontend

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

%I A045691 #17 Jan 29 2022 12:48:53
%S A045691 0,1,1,3,5,11,19,41,77,159,307,625,1231,2481,4921,9883,19689,39455,
%T A045691 78751,157661,315015,630337,1260049,2520723,5040215,10081661,20160841,
%U A045691 40324163,80643405,161291731,322573579,645157041,1290294393,2580608475,5161177495
%N A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.
%C A045691 From _Gus Wiseman_, Jan 22 2022: (Start)
%C A045691 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 A045691   {1}  {2}  {3}    {4}      {5}        {6}
%C A045691             {1,3}  {1,4}    {1,5}      {1,6}
%C A045691             {2,3}  {3,4}    {2,5}      {2,6}
%C A045691                    {1,3,4}  {3,5}      {4,6}
%C A045691                    {2,3,4}  {4,5}      {5,6}
%C A045691                             {1,3,5}    {1,4,6}
%C A045691                             {1,4,5}    {1,5,6}
%C A045691                             {2,3,5}    {2,5,6}
%C A045691                             {3,4,5}    {3,4,6}
%C A045691                             {1,3,4,5}  {3,5,6}
%C A045691                             {2,3,4,5}  {4,5,6}
%C A045691                                        {1,3,4,6}
%C A045691                                        {1,3,5,6}
%C A045691                                        {1,4,5,6}
%C A045691                                        {2,3,4,6}
%C A045691                                        {2,3,5,6}
%C A045691                                        {3,4,5,6}
%C A045691                                        {1,3,4,5,6}
%C A045691                                        {2,3,4,5,6}
%C A045691 (End)
%F A045691 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 A045691 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 A045691 If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:
%Y A045691 - The version with quotients <= 1/2 is A018819, partitions A000929.
%Y A045691 - The version with quotients < 1/2 is A040039, partitions A342098.
%Y A045691 - The version with quotients >= 1/2 is A045690(n+1), partitions A342094.
%Y A045691 - The version with quotients > 1/2 is A045690, partitions A342096.
%Y A045691 - Partitions of this type are counted by A350837, ranked by A350838.
%Y A045691 - Strict partitions of this type are counted by A350840.
%Y A045691 - For differences instead of quotients we have A350842, strict A350844.
%Y A045691 - Partitions not of this type are counted by A350846, ranked by A350845.
%Y A045691 A000740 = relatively prime subsets of {1..n} containing n.
%Y A045691 A002843 = compositions with all adjacent quotients >= 1/2.
%Y A045691 A050291 = double-free subsets of {1..n}.
%Y A045691 A154402 = partitions with all adjacent quotients 2.
%Y A045691 A308546 = double-closed subsets of {1..n}, with maximum: shifted right.
%Y A045691 A323092 = double-free integer partitions, ranked by A320340, strict A120641.
%Y A045691 A326115 = maximal double-free subsets of {1..n}.
%Y A045691 Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.
%K A045691 nonn
%O A045691 0,4
%A A045691 Torsten Sillke (torsten.sillke(AT)lhsystems.com)
%E A045691 More terms from _Sean A. Irvine_, Mar 18 2021