A325096 Number of maximal subsets of {1...n} with no binary carries.

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 12, 13, 14, 15, 15, 20, 25, 27, 32, 34, 36, 37, 42, 44, 46, 47, 49, 50, 51, 52, 52, 67, 82, 87, 102, 107, 112, 114, 129, 134, 139, 141, 146, 148, 150, 151, 166, 171, 176, 178, 183, 185, 187, 188, 193, 195, 197, 198, 200, 201

Offset: 0

Author

Gus Wiseman, Mar 27 2019

Keywords

• nonn
• look

Comments

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

Examples

			The a(1) = 1 through a(9) = 7 maximal subsets:
  {1}  {12}  {3}   {34}   {25}   {16}   {7}    {78}    {69}
             {12}  {124}  {34}   {25}   {16}   {168}   {78}
                          {124}  {34}   {25}   {258}   {168}
                                 {124}  {34}   {348}   {249}
                                        {124}  {1248}  {258}
                                                       {348}
                                                       {1248}
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..8192

Crossrefs

Cf. A000110, A019565, A050315, A080572, A247935, A267610, A267700.

Cf. A325095, A325099, A325100, A325101, A325103, A325104, A325107.

Programs

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]]],{n,0,10}]

Formula

a(2^n - 1) = A000110(n).

Extensions

a(15)-a(61) from Alois P. Heinz, Mar 28 2019