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

0,5

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

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

Example

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}

Maple

f:= proc(n) 2^numboccur(0, convert(n, base, 2))-1 end proc; f(0):= 0:
ListTools:-PartialSums(map(f, [$0..100])); # Robert Israel, Feb 16 2026

Mathematica

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

Crossrefs

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

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

Partial sums of A115378.

Sequence in context: A023846 A338625 A046345 * A004445 A174630 A163875

Adjacent sequences: A325100 A325101 A325102 * A325104 A325105 A325106

Keyword

nonn

Author

Gus Wiseman, Mar 28 2019

Status

approved