A325104 Number of increasing pairs of positive integers up to n with at least one binary carry.

0, 0, 0, 2, 2, 5, 9, 15, 15, 20, 26, 35, 43, 54, 66, 80, 80, 89, 99, 114, 126, 143, 161, 182, 198, 219, 241, 266, 290, 317, 345, 375, 375, 392, 410, 437, 457, 486, 516, 551, 575, 608, 642, 681, 717, 758, 800, 845, 877, 918, 960, 1007, 1051, 1100, 1150, 1203

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.

The version for ordered pairs is A080572.

Examples

			The a(3) = 2 through a(8) = 15 pairs:
  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}
  {2,3}  {2,3}  {1,5}  {1,5}  {1,5}  {1,5}
                {2,3}  {2,3}  {1,7}  {1,7}
                {3,5}  {2,6}  {2,3}  {2,3}
                {4,5}  {3,5}  {2,6}  {2,6}
                       {3,6}  {2,7}  {2,7}
                       {4,5}  {3,5}  {3,5}
                       {4,6}  {3,6}  {3,6}
                       {5,6}  {3,7}  {3,7}
                              {4,5}  {4,5}
                              {4,6}  {4,6}
                              {4,7}  {4,7}
                              {5,6}  {5,6}
                              {5,7}  {5,7}
                              {6,7}  {6,7}
		

Crossrefs

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

Cf. A325094, A325096, A325098, A325102, A325103, 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) = 2 * A080572(n - 2) + n.