Illustration for A0088808-A088812

     ----------------------------------------------------------
     Example: n = 5
     Numbers of double-free and sum-free subsets of {1,2,3,4,5}
     ----------------------------------------------------------


                               +---------------+
              +---------------+| A007865(5)=16 |
              | A085489(5)=22 ||   +-----------+--+
              | A088809(5)=10 ||   | A088810(5)=4 |
        +-----+---------+     ||   |   +----------+---+
        | A050291(5)=20 |     ||   |   | A088811(5)=6 |
        | A088808(5)=12 |     ||   |   |   +----------+---+
        +---------+     |     ||   |   |   | A088812(5)=6 |
                  |     |     ||   |   |   |   +----------+
                  |     |     ||   |   |   |   |
     +------------+-----+-----++---+---+---+---+
     | double-free|  T  |  *  || T | T | F | F |
     +------------+-----+-----++---+---+---+---+
     | sum-free   |  *  |  T  || T | F | T | F |
     +------------+-----+-----++---+---+---+---+
                  |     |     ||   |   |   |   |
      subsets     |     |     ||   |   |   |   |
     +------------+-----+-----++---+---+---+---+
     |     empty  |  y  |  y  || 1 |   |   |   |
     |        {1} |  y  |  y  || 1 |   |   |   |
     |        {2} |  y  |  y  || 1 |   |   |   |
     |        {3} |  y  |  y  || 1 |   |   |   |
     |        {4} |  y  |  y  || 1 |   |   |   |
     |        {5} |  y  |  y  || 1 |   |   |   |
     |      {1,2} | 2   |  y  ||   |   | 1 |   |
     |      {1,3} |  y  |  y  || 1 |   |   |   |
     |      {1,4} |  y  |  y  || 1 |   |   |   |
     |      {1,5} |  y  |  y  || 1 |   |   |   |
     |      {2,3} |  y  |  y  || 1 |   |   |   |
     |      {2,4} | 4   |  y  ||   |   | 1 |   |
     |      {2,5} |  y  |  y  || 1 |   |   |   |
     |      {3,4} |  y  |  y  || 1 |   |   |   |
     |      {3,5} |  y  |  y  || 1 |   |   |   |
     |      {4,5} |  y  |  y  || 1 |   |   |   |
     |    {1,2,3} | 2   | 3   ||   |   |   | 1 |
     |    {1,2,4} | 2 4 |  y  ||   |   | 1 |   |
     |    {1,2,5} | 2   |  y  ||   |   | 1 |   |
     |    {1,3,4} |  y  | 4   ||   | 1 |   |   |
     |    {1,3,5} |  y  |  y  || 1 |   |   |   |
     |    {1,4,5} |  y  | 5   ||   | 1 |   |   |
     |    {2,3,4} | 4   |  y  ||   |   | 1 |   |
     |    {2,3,5} |  y  | 5   ||   | 1 |   |   |
     |    {2,4,5} | 4   |  y  ||   |   | 1 |   |
     |    {3,4,5} |  y  |  y  || 1 |   |   |   |
     |  {1,2,3,4} | 2 4 | 3 4 ||   |   |   | 1 |
     |  {1,2,3,5} | 2   | 3 5 ||   |   |   | 1 |
     |  {1,2,4,5} | 2 4 | 5   ||   |   |   | 1 |
     |  {1,3,4,5} |  y  | 4 5 ||   | 1 |   |   |
     |  {2,3,4,5} | 4   | 5   ||   |   |   | 1 |
     |{1,2,3,4,5} | 2 4 |3 4 5||   |   |   | 1 |
     +------------+-----+-----++---+---+---+---+

     Legend:       y =   y =  || only one 1 per
                 double  sum  || subset-row,
                 -free   -free|| 0s are ommittet
                              ||
                    numbers   || column sum =
                   indicate   || column heads
                  why not "y" ||


     ------------------------------------------------------------------
     Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Oct 19 2003