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A364466 Number of subsets of {1..n} where some element is a difference of two consecutive elements.

%I A364466 #20 Oct 18 2023 04:51:46
%S A364466 0,0,1,2,6,14,34,74,164,345,734,1523,3161,6488,13302,27104,55150,
%T A364466 111823,226443,457586,923721,1862183,3751130,7549354,15184291,
%U A364466 30521675,61322711,123151315,247230601,496158486,995447739,1996668494,4004044396,8027966324,16092990132,32255168125
%N A364466 Number of subsets of {1..n} where some element is a difference of two consecutive elements.
%C A364466 In other words, the elements are not disjoint from their own first differences.
%F A364466 a(n) = 2^n - A364463(n). - _Chai Wah Wu_, Sep 26 2023
%e A364466 The a(0) = 0 through a(5) = 14 subsets:
%e A364466   .  .  {1,2}  {1,2}    {1,2}      {1,2}
%e A364466                {1,2,3}  {2,4}      {2,4}
%e A364466                         {1,2,3}    {1,2,3}
%e A364466                         {1,2,4}    {1,2,4}
%e A364466                         {1,3,4}    {1,2,5}
%e A364466                         {1,2,3,4}  {1,3,4}
%e A364466                                    {1,4,5}
%e A364466                                    {2,3,5}
%e A364466                                    {2,4,5}
%e A364466                                    {1,2,3,4}
%e A364466                                    {1,2,3,5}
%e A364466                                    {1,2,4,5}
%e A364466                                    {1,3,4,5}
%e A364466                                    {1,2,3,4,5}
%t A364466 Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]!={}&]],{n,0,10}]
%o A364466 (Python)
%o A364466 from itertools import combinations
%o A364466 def A364466(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if not set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # _Chai Wah Wu_, Sep 26 2023
%Y A364466 For differences of all pairs we have A093971, complement A196723.
%Y A364466 For partitions we have A363260, complement A364467.
%Y A364466 The complement is counted by A364463.
%Y A364466 For subset-sums instead of differences we have A364534, complement A325864.
%Y A364466 For strict partitions we have A364536, complement A364464.
%Y A364466 A000041 counts integer partitions, strict A000009.
%Y A364466 A008284 counts partitions by length, strict A008289.
%Y A364466 A050291 counts double-free subsets, complement A088808.
%Y A364466 A108917 counts knapsack partitions, strict A275972.
%Y A364466 A325325 counts partitions with all distinct differences, strict A320347.
%Y A364466 Cf. A011782, A212986, A237113, A325877, A325878, A326083, A363225.
%K A364466 nonn
%O A364466 0,4
%A A364466 _Gus Wiseman_, Jul 31 2023
%E A364466 a(21)-a(32) from _Chai Wah Wu_, Sep 26 2023
%E A364466 a(33)-a(35) from _Chai Wah Wu_, Sep 27 2023