%I #17 Jul 27 2019 14:57:51

%S 1,2,4,5,10,12,14,15,30,35,40,42,47,49,51,52,104,119,134,139,154,159,

%T 164,166,181,186,191,193,198,200,202,203,406,458,510,525,577,592,607,

%U 612,664,679,694,699,714,719,724,726,778,793,808,813,828,833,838,840

%N Number of subsets of {1...n} with no binary carries.

%C A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the binary representations of {2,5,8} are:

%C 2 = 10,

%C 5 = 101,

%C 8 = 1000,

%C and since there are no columns with more than one 1, {2,5,8} is counted under a(8).

%H Alois P. Heinz, <a href="/A325095/b325095.txt">Table of n, a(n) for n = 0..16383</a>

%F a(2^n - 1) = A000110(n + 1).

%e The a(1) = 1 through a(7) = 15 subsets:

%e {} {} {} {} {} {} {}

%e {1} {1} {1} {1} {1} {1} {1}

%e {2} {2} {2} {2} {2} {2}

%e {1,2} {3} {3} {3} {3} {3}

%e {1,2} {4} {4} {4} {4}

%e {1,2} {5} {5} {5}

%e {1,4} {1,2} {6} {6}

%e {2,4} {1,4} {1,2} {7}

%e {3,4} {2,4} {1,4} {1,2}

%e {1,2,4} {2,5} {1,6} {1,4}

%e {3,4} {2,4} {1,6}

%e {1,2,4} {2,5} {2,4}

%e {3,4} {2,5}

%e {1,2,4} {3,4}

%e {1,2,4}

%p b:= proc(n, t) option remember; `if`(n=0, 1, b(n-1, t)+

%p `if`(Bits[And](n, t)=0, b(n-1, Bits[Or](n, t)), 0))

%p end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..63); # _Alois P. Heinz_, Mar 28 2019

%t binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Table[Length[Select[Subsets[Range[n]],stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,10}]

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

%Y Cf. A325094, A325096, A325097, A325100, A325103, A325104, A325105.

%K nonn,look

%O 0,2

%A _Gus Wiseman_, Mar 27 2019

%E a(16)-a(55) from _Alois P. Heinz_, Mar 28 2019