This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A325103 #6 Jul 27 2019 14:57:51
%S A325103 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,
%T A325103 81,84,85,88,89,90,90,121,136,151,158,173,180,187,190,205,212,219,222,
%U A325103 229,232,235,236,251,258,265,268,275,278,281,282,289,292,295,296
%N A325103 Number of increasing pairs of positive integers up to n with no binary carries.
%C A325103 A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.
%F A325103 a(n) = A325102(n)/2.
%e A325103 The a(2) = 1 through a(9) = 16 pairs:
%e A325103 {1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2}
%e A325103 {1,4} {1,4} {1,4} {1,4} {1,4} {1,4}
%e A325103 {2,4} {2,4} {1,6} {1,6} {1,6} {1,6}
%e A325103 {3,4} {2,5} {2,4} {2,4} {1,8} {1,8}
%e A325103 {3,4} {2,5} {2,5} {2,4} {2,4}
%e A325103 {3,4} {3,4} {2,5} {2,5}
%e A325103 {2,8} {2,8}
%e A325103 {3,4} {2,9}
%e A325103 {3,8} {3,4}
%e A325103 {4,8} {3,8}
%e A325103 {5,8} {4,8}
%e A325103 {6,8} {4,9}
%e A325103 {7,8} {5,8}
%e A325103 {6,8}
%e A325103 {6,9}
%e A325103 {7,8}
%t A325103 Table[Length[Select[Subsets[Range[n],{2}],Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,30}]
%Y A325103 Cf. A006218, A050315, A080572, A247935, A267610, A267700.
%Y A325103 Cf. A325094, A325096, A325098, A325102, A325104, A325106, A325108, A325123.
%K A325103 nonn
%O A325103 0,5
%A A325103 _Gus Wiseman_, Mar 28 2019