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A365544 Number of subsets of {1..n} containing two distinct elements summing to n.

%I A365544 #17 Aug 30 2024 21:28:21
%S A365544 0,0,0,2,4,14,28,74,148,350,700,1562,3124,6734,13468,28394,56788,
%T A365544 117950,235900,484922,969844,1979054,3958108,8034314,16068628,
%U A365544 32491550,64983100,131029082,262058164,527304974,1054609948,2118785834,4237571668,8503841150,17007682300
%N A365544 Number of subsets of {1..n} containing two distinct elements summing to n.
%H A365544 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-6).
%F A365544 a(n) = 2^n - A068911(n).
%F A365544 From _Alois P. Heinz_, Aug 30 2024: (Start)
%F A365544 G.f.: 2*x^3/((2*x-1)*(3*x^2-1)).
%F A365544 a(n) = 2 * A167762(n-1) for n>=1. (End)
%e A365544 The a(1) = 0 through a(5) = 14 subsets:
%e A365544   .  .  {1,2}    {1,3}      {1,4}
%e A365544         {1,2,3}  {1,2,3}    {2,3}
%e A365544                  {1,3,4}    {1,2,3}
%e A365544                  {1,2,3,4}  {1,2,4}
%e A365544                             {1,3,4}
%e A365544                             {1,4,5}
%e A365544                             {2,3,4}
%e A365544                             {2,3,5}
%e A365544                             {1,2,3,4}
%e A365544                             {1,2,3,5}
%e A365544                             {1,2,4,5}
%e A365544                             {1,3,4,5}
%e A365544                             {2,3,4,5}
%e A365544                             {1,2,3,4,5}
%t A365544 Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],n]&]],{n,0,10}]
%o A365544 (Python)
%o A365544 def A365544(n): return (1<<n) - (3**(n>>1)<<1 if n&1 else 3**(n-1>>1)<<2) if n else 0 # _Chai Wah Wu_, Aug 30 2024
%Y A365544 For strict partitions we have A140106 shifted left.
%Y A365544 The version for partitions is A004526.
%Y A365544 The complement is counted by A068911.
%Y A365544 For all subsets of elements we have A365376.
%Y A365544 Main diagonal k = n of A365541.
%Y A365544 A000009 counts subsets summing to n.
%Y A365544 A007865/A085489/A151897 count certain types of sum-free subsets.
%Y A365544 A093971/A088809/A364534 count certain types of sum-full subsets.
%Y A365544 A365381 counts subsets with a subset summing to k.
%Y A365544 Cf. A008967, A095944, A167762, A238628, A288728, A326083, A364272, A365377.
%K A365544 nonn,easy
%O A365544 0,4
%A A365544 _Gus Wiseman_, Sep 20 2023