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%I #13 Jul 27 2019 14:57:51
%S 1,2,3,7,8,20,48,112,113,325,777,1737,3709,7741,15869,32253,32254,
%T 96538,225798,485702,1006338,2049602,4137346,8315266,16697102,
%U 33465934,67007886,134100366,268301518,536720590,1073575118,2147316942,2147316943,6441886323
%N Number of binary carry-connected subsets of {1...n}.
%C A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. A subset is binary carry-connected if the graph whose vertices are the elements and whose edges are binary carries is connected.
%H Alois P. Heinz, <a href="/A325105/b325105.txt">Table of n, a(n) for n = 0..1023</a>
%F a(n) = A306297(n,0) + A306297(n,1). - _Alois P. Heinz_, Mar 31 2019
%e The a(0) = 1 through a(4) = 8 subsets:
%e {} {} {} {} {}
%e {1} {1} {1} {1}
%e {2} {2} {2}
%e {3} {3}
%e {1,3} {4}
%e {2,3} {1,3}
%e {1,2,3} {2,3}
%e {1,2,3}
%p h:= proc(n, s) local i, m; m:= n;
%p for i in s do m:= Bits[Or](m, i) od; {m}
%p end:
%p g:= (n, s)-> (w-> `if`(w={}, s union {n}, s minus w union
%p h(n, w)))(select(x-> Bits[And](n, x)>0, s)):
%p b:= proc(n, s) option remember; `if`(n=0,
%p `if`(nops(s)>1, 0, 1), b(n-1, s)+b(n-1, g(n, s)))
%p end:
%p a:= n-> b(n, {}):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Mar 31 2019
%t binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t Table[Length[Select[Subsets[Range[n]],Length[csm[binpos/@#]]<=1&]],{n,0,10}]
%Y Cf. A019565, A080572, A247935, A304714, A304716, A305078.
%Y Cf. A325095, A325098, A325099, A325104, A325107, A325118, A325119.
%Y Partial sums of A306299.
%K nonn
%O 0,2
%A _Gus Wiseman_, Mar 28 2019
%E a(16)-a(33) from _Alois P. Heinz_, Mar 31 2019
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