OFFSET
1,3
COMMENTS
In sumset notation, number of subsets A of {1,...,n} such that the intersection of A and 2A is nonempty.
A variation of binary sum-full sets where parts can be re-used, this sequence counts subsets of {1..n} containing a part equal to the sum of two other (possibly equal) parts. The complement is counted by A007865. The non-binary version is A364914. For non-re-usable parts we have A088809. - Gus Wiseman, Aug 14 2023
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 1..88
Eric Weisstein's World of Mathematics, Sum-Free Set
FORMULA
a(n) = 2^n - A007865(n).
EXAMPLE
The a(1) = 0 through a(5) = 16 subsets:
. {1,2} {1,2} {1,2} {1,2}
{1,2,3} {2,4} {2,4}
{1,2,3} {1,2,3}
{1,2,4} {1,2,4}
{1,3,4} {1,2,5}
{2,3,4} {1,3,4}
{1,2,3,4} {1,4,5}
{2,3,4}
{2,3,5}
{2,4,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], Intersection[#, Total/@Tuples[#, 2]]!={}&]], {n, 0, 10}] (* Gus Wiseman, Aug 14 2023 *)
CROSSREFS
The complement is counted by A007865.
The version without re-usable parts is A088809 (differences A364756), complement A085489 (differences A364755).
The version for partitions is A363225:
- ranks A364348,
- strict A363226,
- non-binary A364839,
- without re-usable parts A237113,
- non-binary without re-usable parts A237668.
The complement for partitions is A364345:
- ranks A364347,
- strict A364346,
- non-binary A364350,
- without re-usable parts A236912,
- non-binary without re-usable parts A237667.
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 20 2004
EXTENSIONS
Terms a(31) and beyond from Fausto A. C. Cariboni, Oct 01 2020
STATUS
approved