A325103 Number of increasing pairs of positive integers up to n with no binary carries.

0, 0, 1, 1, 4, 5, 6, 6, 13, 16, 19, 20, 23, 24, 25, 25, 40, 47, 54, 57, 64, 67, 70, 71, 78, 81, 84, 85, 88, 89, 90, 90, 121, 136, 151, 158, 173, 180, 187, 190, 205, 212, 219, 222, 229, 232, 235, 236, 251, 258, 265, 268, 275, 278, 281, 282, 289, 292, 295, 296

Offset: 0

Author

Gus Wiseman, Mar 28 2019

Keywords

• nonn

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(2) = 1 through a(9) = 16 pairs:
  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}
                {1,4}  {1,4}  {1,4}  {1,4}  {1,4}  {1,4}
                {2,4}  {2,4}  {1,6}  {1,6}  {1,6}  {1,6}
                {3,4}  {2,5}  {2,4}  {2,4}  {1,8}  {1,8}
                       {3,4}  {2,5}  {2,5}  {2,4}  {2,4}
                              {3,4}  {3,4}  {2,5}  {2,5}
                                            {2,8}  {2,8}
                                            {3,4}  {2,9}
                                            {3,8}  {3,4}
                                            {4,8}  {3,8}
                                            {5,8}  {4,8}
                                            {6,8}  {4,9}
                                            {7,8}  {5,8}
                                                   {6,8}
                                                   {6,9}
                                                   {7,8}
		

Crossrefs

Cf. A006218, A050315, A080572, A247935, A267610, A267700.

Cf. A325094, A325096, A325098, A325102, A325104, A325106, A325108, A325123.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n],{2}],Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,30}]

Formula

a(n) = A325102(n)/2.