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A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.

%I A365069 #11 Dec 13 2024 09:37:33
%S A365069 0,0,0,1,2,7,17,41,88,201,418,892,1838,3798,7716,15740
%N A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.
%C A365069 The complement is counted by A365071. The binary case is A364756. Allowing elements to be re-used gives A365070. A version for partitions (but not requiring n) is A237668.
%H A365069 Steven R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.
%F A365069 a(n) = 2^(n-1) - A365070(n).
%F A365069 First differences of A364534.
%e A365069 The subset {2,4,6} has 6 = 4 + 2 so is counted under a(6).
%e A365069 The subset {1,2,4,7} has 7 = 4 + 2 + 1 so is counted under a(7).
%e A365069 The subset {1,4,5,8} has 5 = 4 + 1 so is counted under a(8).
%e A365069 The a(0) = 0 through a(6) = 17 subsets:
%e A365069   .  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
%e A365069                     {1,2,3,4}  {2,3,5}      {2,4,6}
%e A365069                                {1,2,3,5}    {1,2,3,6}
%e A365069                                {1,2,4,5}    {1,2,4,6}
%e A365069                                {1,3,4,5}    {1,2,5,6}
%e A365069                                {2,3,4,5}    {1,3,4,6}
%e A365069                                {1,2,3,4,5}  {1,3,5,6}
%e A365069                                             {1,4,5,6}
%e A365069                                             {2,3,4,6}
%e A365069                                             {2,3,5,6}
%e A365069                                             {2,4,5,6}
%e A365069                                             {1,2,3,4,6}
%e A365069                                             {1,2,3,5,6}
%e A365069                                             {1,2,4,5,6}
%e A365069                                             {1,3,4,5,6}
%e A365069                                             {2,3,4,5,6}
%e A365069                                             {1,2,3,4,5,6}
%t A365069 Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#, Total/@Subsets[#, {2,Length[#]}]]!={}&]],{n,0,10}]
%Y A365069 The complement w/ re-usable parts is A288728, first differences of A007865.
%Y A365069 First differences of A364534.
%Y A365069 The binary complement is A364755, first differences of A085489.
%Y A365069 The binary version is A364756, first differences of A088809.
%Y A365069 The version with re-usable parts is A365070, first differences of A093971.
%Y A365069 The complement is counted by A365071, first differences of A151897.
%Y A365069 A124506 counts nonnegative combination-free subsets, differences of A326083.
%Y A365069 A365046 counts nonnegative combination-full subsets, differences of A364914.
%Y A365069 For partitions: A108917, A236912, A237113, A237668, A364532, A364913.
%Y A365069 Strict partitions: A116861, A364272, A364349, A364350, A364839, A364916.
%Y A365069 Cf. A050291, A326080, A363226, A364346, A364348, A364670.
%K A365069 nonn
%O A365069 0,5
%A A365069 _Gus Wiseman_, Aug 26 2023