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A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.

%I A365046 #8 Dec 13 2024 09:39:24
%S A365046 0,0,1,2,6,11,28,53,118,235,490,973,2008,3990,8089,16184,32563,65071,
%T A365046 130667,261183,523388,1046748,2095239,4190208,8385030,16768943,
%U A365046 33546257,67092732,134201461,268400553,536839090,1073670970,2147414967,4294829905,8589793931
%N A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.
%C A365046 Includes all subsets containing both 1 and n.
%H A365046 Steven R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.
%F A365046 a(n+1) = 2^n - A124506(n).
%e A365046 The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
%e A365046 The a(0) = 0 through a(5) = 11 subsets:
%e A365046   .  .  {1,2}  {1,3}    {1,4}      {1,5}
%e A365046                {1,2,3}  {2,4}      {1,2,5}
%e A365046                         {1,2,4}    {1,3,5}
%e A365046                         {1,3,4}    {1,4,5}
%e A365046                         {2,3,4}    {2,3,5}
%e A365046                         {1,2,3,4}  {2,4,5}
%e A365046                                    {1,2,3,5}
%e A365046                                    {1,2,4,5}
%e A365046                                    {1,3,4,5}
%e A365046                                    {2,3,4,5}
%e A365046                                    {1,2,3,4,5}
%t A365046 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365046 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combs[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]
%Y A365046 The complement is A124506, first differences of A326083.
%Y A365046 The binary complement is A288728, first differences of A007865.
%Y A365046 First differences of A364914.
%Y A365046 The positive version is A365042, first differences of A365043.
%Y A365046 The positive complement is counted by A365045, first differences of A365044.
%Y A365046 Without re-usable parts we have A365069, first differences of A364534.
%Y A365046 The binary version is A365070, first differences of A093971.
%Y A365046 A364350 counts combination-free strict partitions, complement A364839.
%Y A365046 A085489 and A364755 count subsets without the sum of two distinct elements.
%Y A365046 A088809 and A364756 count subsets with the sum of two distinct elements.
%Y A365046 A364913 counts combination-full partitions.
%Y A365046 Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.
%K A365046 nonn
%O A365046 0,4
%A A365046 _Gus Wiseman_, Aug 24 2023