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

1, 2, 3, 6, 8, 16, 24, 48, 60, 120, 180, 360, 480, 960, 1440, 2880, 3456, 6912, 10368, 20736, 27648, 55296, 82944, 165888, 207360, 414720, 622080, 1244160, 1658880, 3317760, 4976640, 9953280, 11612160, 23224320, 34836480, 69672960, 92897280

Offset

0,2

Comments

These are subsets containing twice any element whose double is <= n.

Also the number of subsets of {1..n} containing half of every element that is even. For example, the a(6) = 24 subsets are:

{} {1} {1,2} {1,2,3} {1,2,3,4} {1,2,3,4,5} {1,2,3,4,5,6}

{3} {1,3} {1,2,4} {1,2,3,5} {1,2,3,4,6}

{5} {1,5} {1,2,5} {1,2,3,6} {1,2,3,5,6}

{3,5} {1,3,5} {1,2,4,5}

{3,6} {1,3,6} {1,3,5,6}

{3,5,6}

Links

Charlie Neder, Table of n, a(n) for n = 0..500

Formula

From Charlie Neder, Jun 10 2019: (Start)

a(n) = Product_{k < n/2} (2 + floor(log_2(n/(2k+1)))).

a(0) = 1, a(n) = a(n-1) * (1 + 1/A001511(n)). (End)

Example

The a(6) = 24 subsets:
  {}  {4}  {2,4}  {1,2,4}  {1,2,4,5}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {5}  {3,6}  {2,4,5}  {1,2,4,6}  {1,2,4,5,6}
      {6}  {4,5}  {2,4,6}  {2,3,4,6}  {2,3,4,5,6}
           {4,6}  {3,4,6}  {2,4,5,6}
           {5,6}  {3,5,6}  {3,4,5,6}
                  {4,5,6}

Mathematica

Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Select[2*#, #<=n&]]&]], {n, 0, 10}]

Crossrefs

Cf. A007865, A050291, A103580, A120641, A320340, A323092, A325864, A326020, A326076, A326083, A326115.

Sequence in context: A334269 A029867 A056348 * A324737 A057574 A198296

Adjacent sequences: A308543 A308544 A308545 * A308547 A308548 A308549

Keyword

nonn

Author

Gus Wiseman, Jun 06 2019

Extensions

a(21)-a(36) from Charlie Neder, Jun 10 2019

Status

approved