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A325105 Number of binary carry-connected subsets of {1...n}. 12
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 (list; graph; refs; listen; history; text; internal format)
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
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
Partial sums of A306299.
Sequence in context: A247843 A181658 A251541 * A276032 A114281 A137823
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 28 2019
EXTENSIONS
a(16)-a(33) from Alois P. Heinz, Mar 31 2019
STATUS
approved

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Last modified April 17 07:54 EDT 2024. Contains 371756 sequences. (Running on oeis4.)