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