A367400 Number of subsets of {1..n} whose cardinality is not the sum of two distinct elements.

1, 2, 4, 7, 13, 25, 47, 88, 166, 313, 589, 1109, 2089, 3934, 7408, 13951, 26273, 49477, 93175, 175468, 330442, 622289, 1171897, 2206921, 4156081, 7826746, 14739356, 27757207, 52272469, 98439697, 185381983, 349112000, 657448942, 1238110153

Offset: 0

Author

Gus Wiseman, Nov 21 2023

Keywords

• nonn
• more

Examples

			The set s = {1,2,3,6,7,8} has the following sums of pairs of distinct elements: {3,4,5,7,8,9,10,11,13,14,15}. This does not include 6, so s is counted under a(8).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,3,4}
                         {2,3,4}
		

Crossrefs

The version containing n appears to be A112575.

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.

sum-full sum-free comb-full comb-free semi-full semi-free

-----------------------------------------------------------

partitions: A367212 A367213 A367218 A367219 A367394 A367398

strict: A367214 A367215 A367220 A367221 A367395 A367399

subsets: A367216 A367217 A367222 A367223 A367396 A367400*

ranks: A367224 A367225 A367226 A367227 A367397 A367401

A002865 counts partitions whose length is a part, complement A229816.

A364534 counts sum-full subsets.

A088809 and A093971 count subsets containing semi-sums.

A236912 counts partitions with no semi-sum of the parts, ranks A364461.

A366738 counts semi-sums of partitions, strict A366741.

Triangles:

A365381 counts subsets with a subset summing to k, complement A366320.

A365541 counts subsets with a semi-sum k.

A367404 counts partitions with a semi-sum k, strict A367405.

Cf. A237113, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366131, A366740.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n,0,10}]

Python

from itertools import combinations
def A367400(n): return (n*(n+1)>>1)+1+sum(1 for k in range(3,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if not any({a,k-a}<=w for a in range(1,k+1>>1))) # Chai Wah Wu, Nov 21 2023

Formula

Conjectures from Chai Wah Wu, Nov 21 2023: (Start)

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) for n > 3.

G.f.: (-x^3 + x^2 + 1)/(x^4 - 2*x^3 + x^2 - 2*x + 1). (End)

Extensions

a(18)-a(33) from Chai Wah Wu, Nov 21 2023