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Date: 30-10-2019
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Date: 22-7-2020
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Date: 10-10-2020
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A set of positive integers is double-free if, for any integer , the set
(or equivalently,
implies
). For example, of the subsets of
, the sets
,
,
,
,
, and
are double-free, while
and
are not.
The number of double-free subsets of
can be computed using
and the recurrence relation
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(1) |
where is a Fibonacci number, 1, 1, 2, 3, 5, 8, ... (OEIS A000045), and
is the binary carry sequence giving the number of trailing 0s in the binary representation of
. For
, 2, ...,
is given by 0, 1, 0, 2, 0, 1, 3, 0, 1, ... (OEIS A007814), while the corresponding
are 2, 3, 6, 10, 20, 30, 60, 96, 192, ... (OEIS A050291).
Define
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(2) |
where is the cardinal number of (number of members in)
. Then for
, 2, ...,
is given by 1, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, ... (OEIS A050292). An explicit formula for
is given by
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(3) |
where
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(4) |
if the characteristic function of (defined above), and the first few values of
are 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ... (OEIS A035263). A simple recurrence relation for
is given by
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(5) |
with (Wang 1989), where
is the floor function and
is the ceiling function. An asymptotic formula for
is given by
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(6) |
(Wang 1989).
REFERENCES:
Finch, S. R. "Triple-Free Set Constants." §2.26 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 183-185, 2003.
Sloane, N. J. A. Sequences A000045/M0692, A007814, A035263, A050291 and A050292 in "The On-Line Encyclopedia of Integer Sequences."
Wang, E. T. H. "On Double-Free Sets of Integers." Ars Combin. 28, 97-100, 1989.
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