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%I #40 Jul 08 2019 07:44:06
%S 1,2,3,6,10,20,30,60,96,192,288,576,960,1920,2880,5760,9360,18720,
%T 28080,56160,93600,187200,280800,561600,898560,1797120,2695680,
%U 5391360,8985600,17971200,26956800,53913600,87091200,174182400,261273600,522547200,870912000
%N Number of double-free subsets of {1, 2, ..., n}.
%C A set is double-free if it does not contain both x and 2x.
%C So these are equally "half-free" subsets. - _Gus Wiseman_, Jul 08 2019
%D Wang, E. T. H. ``On Double-Free Sets of Integers.'' Ars Combin. 28, 97-100, 1989.
%H Alois P. Heinz, <a href="/A050291/b050291.txt">Table of n, a(n) for n = 0..4030</a> (terms n = 1..400 from T. D. Noe)
%H Steven R. Finch, <a href="/FinchTriple.html">Triple-Free Sets of Integers</a> [From Steven Finch, Apr 20 2019]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Double-FreeSet.html">Double-Free Set.</a>
%F a(n) = a(n-1)*Fibonacci(b(2n)+2)/Fibonacci(b(2n)+1), Fibonacci = A000045, b = A007814.
%F a(n) = 2^n - A088808(n). - _Reinhard Zumkeller_, Oct 19 2003
%e From _Gus Wiseman_, Jul 08 2019: (Start)
%e The a(0) = 1 through a(5) = 20 double-free subsets:
%e {} {} {} {} {} {}
%e {1} {1} {1} {1} {1}
%e {2} {2} {2} {2}
%e {3} {3} {3}
%e {1,3} {4} {4}
%e {2,3} {1,3} {5}
%e {1,4} {1,3}
%e {2,3} {1,4}
%e {3,4} {1,5}
%e {1,3,4} {2,3}
%e {2,5}
%e {3,4}
%e {3,5}
%e {4,5}
%e {1,3,4}
%e {1,3,5}
%e {1,4,5}
%e {2,3,5}
%e {3,4,5}
%e {1,3,4,5}
%e (End)
%p a:= proc(n) option remember; `if`(n=0, 1, (F-> (p-> a(n-1)*F(p+3)
%p /F(p+2))(padic[ordp](n, 2)))(j-> (<<0|1>, <1|1>>^j)[1, 2]))
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 16 2019
%t 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-François Alcover_, Oct 10 2012, from first formula *)
%t Table[Length[Select[Subsets[Range[n]],Intersection[#,#/2]=={}&]],{n,0,10}] (* _Gus Wiseman_, Jul 08 2019 *)
%o (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
%Y Cf. A000045, A007814, A050292-A050296.
%Y Cf. A007865, A103580, A120641, A308546, A320340, A323092, A326083, A326115.
%K nonn,easy,nice
%O 0,2
%A _Eric W. Weisstein_
%E Extended with formula by _Christian G. Bower_, Sep 15 1999
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 16 2019
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