A326080 - OEIS

%I #13 Aug 30 2019 21:47:48

%S 1,2,4,7,12,19,31,47,73,110,168,247,375,546,817,1193,1769,2552,3791,

%T 5445,8012,11517,16899,24144,35391,50427,73614,104984,152656,216802,

%U 315689,447473,648813,920163,1332991,1884735,2728020,3853437,5568644,7868096,11347437

%N Number of subsets of {1..n} containing the sum of every subset whose sum is <= n.

%C Equivalently, a(n) is the number of subsets of {1..n} containing the sum of any two distinct elements whose sum is <= n.

%C The summands must be distinct. The case where they are allowed to be equal is A326083.

%C If A151897 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).

%e The a(0) = 1 through a(5) = 19 subsets:

%e   {}  {}   {}     {}       {}         {}

%e       {1}  {1}    {1}      {1}        {1}

%e            {2}    {2}      {2}        {2}

%e            {1,2}  {3}      {3}        {3}

%e                   {1,3}    {4}        {4}

%e                   {2,3}    {1,4}      {5}

%e                   {1,2,3}  {2,3}      {1,5}

%e                            {2,4}      {2,4}

%e                            {3,4}      {2,5}

%e                            {1,3,4}    {3,4}

%e                            {2,3,4}    {3,5}

%e                            {1,2,3,4}  {4,5}

%e                                       {1,4,5}

%e                                       {2,3,5}

%e                                       {2,4,5}

%e                                       {3,4,5}

%e                                       {1,3,4,5}

%e                                       {2,3,4,5}

%e                                       {1,2,3,4,5}

%e The a(6) = 31 subsets:

%e   {}  {1}  {1,6}  {1,5,6}  {1,4,5,6}  {1,3,4,5,6}  {1,2,3,4,5,6}

%e       {2}  {2,5}  {2,3,5}  {2,3,5,6}  {2,3,4,5,6}

%e       {3}  {2,6}  {2,4,6}  {2,4,5,6}

%e       {4}  {3,4}  {2,5,6}  {3,4,5,6}

%e       {5}  {3,5}  {3,4,5}

%e       {6}  {3,6}  {3,4,6}

%e            {4,5}  {3,5,6}

%e            {4,6}  {4,5,6}

%e            {5,6}

%t Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[Plus@@@Subsets[#,{2}],#<=n&]]&]],{n,0,10}]

%o (PARI)

%o a(n)={

%o    my(recurse(k, b)=

%o       if( k > n, 1,

%o           my(t=self()(k + 1, b + (1<<k)));

%o           for(i=1, (k-1)\2, if(bittest(b, i) && bittest(b, k-i), return(t)));

%o           t + self()(k + 1, b) )

%o    );

%o    recurse(1, 0);

%o } \\ _Andrew Howroyd_, Aug 30 2019

%Y Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326083.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 05 2019

%E Terms a(21) and beyond from _Andrew Howroyd_, Aug 30 2019