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A050291 Number of double-free subsets of {1, 2, ..., n}. 18
1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 288, 576, 960, 1920, 2880, 5760, 9360, 18720, 28080, 56160, 93600, 187200, 280800, 561600, 898560, 1797120, 2695680, 5391360, 8985600, 17971200, 26956800, 53913600, 87091200, 174182400, 261273600, 522547200, 870912000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A set is double-free if it does not contain both x and 2x.

So these are equally "half-free" subsets. - Gus Wiseman, Jul 08 2019

REFERENCES

Wang, E. T. H. ``On Double-Free Sets of Integers.'' Ars Combin. 28, 97-100, 1989.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..4030 (terms n = 1..400 from T. D. Noe)

Steven R. Finch, Triple-Free Sets of Integers [From Steven Finch, Apr 20 2019]

Eric Weisstein's World of Mathematics, Double-Free Set.

FORMULA

a(n) = a(n-1)*Fibonacci(b(2n)+2)/Fibonacci(b(2n)+1), Fibonacci = A000045, b = A007814.

a(n) = 2^n - A088808(n). - Reinhard Zumkeller, Oct 19 2003

EXAMPLE

From Gus Wiseman, Jul 08 2019: (Start)

The a(0) = 1 through a(5) = 20 double-free subsets:

  {}  {}   {}   {}     {}       {}

      {1}  {1}  {1}    {1}      {1}

           {2}  {2}    {2}      {2}

                {3}    {3}      {3}

                {1,3}  {4}      {4}

                {2,3}  {1,3}    {5}

                       {1,4}    {1,3}

                       {2,3}    {1,4}

                       {3,4}    {1,5}

                       {1,3,4}  {2,3}

                                {2,5}

                                {3,4}

                                {3,5}

                                {4,5}

                                {1,3,4}

                                {1,3,5}

                                {1,4,5}

                                {2,3,5}

                                {3,4,5}

                                {1,3,4,5}

(End)

MAPLE

a:= proc(n) option remember; `if`(n=0, 1, (F-> (p-> a(n-1)*F(p+3)

      /F(p+2))(padic[ordp](n, 2)))(j-> (<<0|1>, <1|1>>^j)[1, 2]))

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Jan 16 2019

MATHEMATICA

a[n_] := a[n] = (b = IntegerExponent[2n, 2]; a[n-1]*Fibonacci[b+2]/Fibonacci[b+1]); a[1]=2; Table[a[n], {n, 1, 34}] (* Jean-Fran├žois Alcover, Oct 10 2012, from first formula *)

Table[Length[Select[Subsets[Range[n]], Intersection[#, #/2]=={}&]], {n, 0, 10}] (* Gus Wiseman, Jul 08 2019 *)

PROG

(PARI) first(n)=my(v=vector(n)); v[1]=2; for(k=2, n, v[k]=v[k-1]*fibonacci(valuation(k, 2)+3)/fibonacci(valuation(k, 2)+2)); v \\ Charles R Greathouse IV, Feb 07 2017

CROSSREFS

Cf. A000045, A007814, A050292-A050296.

Cf. A007865, A103580, A120641, A308546, A320340, A323092, A326083, A326115.

Sequence in context: A247162 A001678 A113292 * A324739 A214002 A305889

Adjacent sequences:  A050288 A050289 A050290 * A050292 A050293 A050294

KEYWORD

nonn,easy,nice

AUTHOR

Eric W. Weisstein

EXTENSIONS

Extended with formula by Christian G. Bower, Sep 15 1999

a(0)=1 prepended by Alois P. Heinz, Jan 16 2019

STATUS

approved

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Last modified April 9 14:04 EDT 2020. Contains 333353 sequences. (Running on oeis4.)