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A308546 Number of double-closed subsets of {1..n}.

%I A308546 #17 Jun 11 2019 00:28:31
%S A308546 1,2,3,6,8,16,24,48,60,120,180,360,480,960,1440,2880,3456,6912,10368,
%T A308546 20736,27648,55296,82944,165888,207360,414720,622080,1244160,1658880,
%U A308546 3317760,4976640,9953280,11612160,23224320,34836480,69672960,92897280
%N A308546 Number of double-closed subsets of {1..n}.
%C A308546 These are subsets containing twice any element whose double is <= n.
%C A308546 Also the number of subsets of {1..n} containing half of every element that is even. For example, the a(6) = 24 subsets are:
%C A308546   {}  {1}  {1,2}  {1,2,3}  {1,2,3,4}  {1,2,3,4,5}  {1,2,3,4,5,6}
%C A308546       {3}  {1,3}  {1,2,4}  {1,2,3,5}  {1,2,3,4,6}
%C A308546       {5}  {1,5}  {1,2,5}  {1,2,3,6}  {1,2,3,5,6}
%C A308546            {3,5}  {1,3,5}  {1,2,4,5}
%C A308546            {3,6}  {1,3,6}  {1,3,5,6}
%C A308546                   {3,5,6}
%H A308546 Charlie Neder, <a href="/A308546/b308546.txt">Table of n, a(n) for n = 0..500</a>
%F A308546 From _Charlie Neder_, Jun 10 2019: (Start)
%F A308546 a(n) = Product_{k < n/2} (2 + floor(log_2(n/(2k+1)))).
%F A308546 a(0) = 1, a(n) = a(n-1) * (1 + 1/A001511(n)). (End)
%e A308546 The a(6) = 24 subsets:
%e A308546   {}  {4}  {2,4}  {1,2,4}  {1,2,4,5}  {1,2,3,4,6}  {1,2,3,4,5,6}
%e A308546       {5}  {3,6}  {2,4,5}  {1,2,4,6}  {1,2,4,5,6}
%e A308546       {6}  {4,5}  {2,4,6}  {2,3,4,6}  {2,3,4,5,6}
%e A308546            {4,6}  {3,4,6}  {2,4,5,6}
%e A308546            {5,6}  {3,5,6}  {3,4,5,6}
%e A308546                   {4,5,6}
%t A308546 Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[2*#,#<=n&]]&]],{n,0,10}]
%Y A308546 Cf. A007865, A050291, A103580, A120641, A320340, A323092, A325864, A326020, A326076, A326083, A326115.
%K A308546 nonn
%O A308546 0,2
%A A308546 _Gus Wiseman_, Jun 06 2019
%E A308546 a(21)-a(36) from _Charlie Neder_, Jun 10 2019