A325105 Number of binary carry-connected subsets of {1...n}.

1, 2, 3, 7, 8, 20, 48, 112, 113, 325, 777, 1737, 3709, 7741, 15869, 32253, 32254, 96538, 225798, 485702, 1006338, 2049602, 4137346, 8315266, 16697102, 33465934, 67007886, 134100366, 268301518, 536720590, 1073575118, 2147316942, 2147316943, 6441886323

Offset

0,2

Comments

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1023

Formula

a(n) = A306297(n,0) + A306297(n,1). - Alois P. Heinz, Mar 31 2019

Example

The a(0) = 1 through a(4) = 8 subsets:
  {}  {}   {}   {}       {}
      {1}  {1}  {1}      {1}
           {2}  {2}      {2}
                {3}      {3}
                {1,3}    {4}
                {2,3}    {1,3}
                {1,2,3}  {2,3}
                         {1,2,3}

Maple

h:= proc(n, s) local i, m; m:= n;
      for i in s do m:= Bits[Or](m, i) od; {m}
    end:
g:= (n, s)-> (w-> `if`(w={}, s union {n}, s minus w union
              h(n, w)))(select(x-> Bits[And](n, x)>0, s)):
b:= proc(n, s) option remember; `if`(n=0,
      `if`(nops(s)>1, 0, 1), b(n-1, s)+b(n-1, g(n, s)))
    end:
a:= n-> b(n, {}):
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 31 2019

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
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]]]]]]]]];
Table[Length[Select[Subsets[Range[n]], Length[csm[binpos/@#]]<=1&]], {n, 0, 10}]

Crossrefs

Cf. A019565, A080572, A247935, A304714, A304716, A305078.

Cf. A325095, A325098, A325099, A325104, A325107, A325118, A325119.

Partial sums of A306299.

Sequence in context: A247843 A181658 A251541 * A276032 A114281 A137823

Adjacent sequences: A325102 A325103 A325104 * A325106 A325107 A325108

Keyword

nonn

Author

Gus Wiseman, Mar 28 2019

Extensions

a(16)-a(33) from Alois P. Heinz, Mar 31 2019

Status

approved