OFFSET
0,4
COMMENTS
From Gus Wiseman, Jan 22 2022: (Start)
Also the number of subsets of {1..n} containing n but without adjacent elements of quotient 1/2. The Heinz numbers of these sets are a subset of the squarefree terms of A320340. For example, the a(1) = 1 through a(6) = 19 subsets are:
{1} {2} {3} {4} {5} {6}
{1,3} {1,4} {1,5} {1,6}
{2,3} {3,4} {2,5} {2,6}
{1,3,4} {3,5} {4,6}
{2,3,4} {4,5} {5,6}
{1,3,5} {1,4,6}
{1,4,5} {1,5,6}
{2,3,5} {2,5,6}
{3,4,5} {3,4,6}
{1,3,4,5} {3,5,6}
{2,3,4,5} {4,5,6}
{1,3,4,6}
{1,3,5,6}
{1,4,5,6}
{2,3,4,6}
{2,3,5,6}
{3,4,5,6}
{1,3,4,5,6}
{2,3,4,5,6}
(End)
FORMULA
a(2*n-1) = 2*a(2*n-2) - a(n) for n >= 2; a(2*n) = 2*a(2*n-1) + a(n) for n >= 2.
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&And@@Table[#[[i-1]]/#[[i]]!=1/2, {i, 2, Length[#]}]&]], {n, 0, 15}] (* Gus Wiseman, Jan 22 2022 *)
CROSSREFS
If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:
- Strict partitions of this type are counted by A350840.
A000740 = relatively prime subsets of {1..n} containing n.
A002843 = compositions with all adjacent quotients >= 1/2.
A050291 = double-free subsets of {1..n}.
A154402 = partitions with all adjacent quotients 2.
A308546 = double-closed subsets of {1..n}, with maximum: shifted right.
A326115 = maximal double-free subsets of {1..n}.
KEYWORD
nonn
AUTHOR
Torsten Sillke (torsten.sillke(AT)lhsystems.com)
EXTENSIONS
More terms from Sean A. Irvine, Mar 18 2021
STATUS
approved