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A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

%I A365042 #15 Dec 13 2024 09:37:37
%S A365042 0,0,1,2,4,5,9,11,17,21,29,36,50,60,78,95,123,147,185,221,274,325,399,
%T A365042 472,574,672,810,945,1131,1316,1557,1812,2137,2462,2892,3322,3881,
%U A365042 4460,5176,5916,6846,7817,8993,10250,11765,13333,15280,17308,19731,22306
%N A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.
%C A365042 Sets of this type may be called "positive combination-full".
%C A365042 Also subsets of {1..n} containing n whose greatest element can be written as a positive linear combination of the others.
%H A365042 Steven R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.
%F A365042 a(n) = A088314(n) - 1.
%e A365042 The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
%e A365042 The a(0) = 0 through a(7) = 11 subsets:
%e A365042   .  .  {1,2}  {1,3}    {1,4}    {1,5}    {1,6}      {1,7}
%e A365042                {1,2,3}  {2,4}    {1,2,5}  {2,6}      {1,2,7}
%e A365042                         {1,2,4}  {1,3,5}  {3,6}      {1,3,7}
%e A365042                         {1,3,4}  {1,4,5}  {1,2,6}    {1,4,7}
%e A365042                                  {2,3,5}  {1,3,6}    {1,5,7}
%e A365042                                           {1,4,6}    {1,6,7}
%e A365042                                           {1,5,6}    {2,3,7}
%e A365042                                           {2,4,6}    {2,5,7}
%e A365042                                           {1,2,3,6}  {3,4,7}
%e A365042                                                      {1,2,3,7}
%e A365042                                                      {1,2,4,7}
%t A365042 combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365042 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combp[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]
%Y A365042 The nonnegative complement is A124506, first differences of A326083.
%Y A365042 The binary complement is A288728, first differences of A007865.
%Y A365042 First differences of A365043.
%Y A365042 The complement is counted by A365045, first differences of A365044.
%Y A365042 The nonnegative version is A365046, first differences of A364914.
%Y A365042 Without re-usable parts we have A365069, first differences of A364534.
%Y A365042 The binary version is A365070, first differences of A093971.
%Y A365042 A085489 and A364755 count subsets with no sum of two distinct elements.
%Y A365042 A088314 counts sets that can be linearly combined to obtain n.
%Y A365042 A088809 and A364756 count subsets with some sum of two distinct elements.
%Y A365042 A364350 counts combination-free strict partitions, complement A364839.
%Y A365042 A364913 counts combination-full partitions.
%Y A365042 Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.
%K A365042 nonn
%O A365042 0,4
%A A365042 _Gus Wiseman_, Aug 23 2023