Notes on A121194
David Applegate
Aug 17 2006

 Using * to indicate a curved side (without specifying in/out),
 and ^, v, <, and > to indicate curved sides, the rules for A121194 mean:

   +---+   +***+   are not allowed, because every 1x2 has one curved
   |   |   |   |   long side.
   +---+   +***+

   +---+    is not allowed, because all curved edges must be on
   |   |    the boundary
   +***+
   |   |
   +---+

   +---+***+  +***+-+   are not allowed, because each
   |   |   |  |   | *   polyomino edge must be either all
   +***+---+  +---+ *   curved, or all straight
                  | *
                  +-+

   +***+    is not allowed, because if two cells meet on a
   |   |    side of length 2, they must share the whole side.
   +-+-+-+
     |   |
     +***+

   +^^^+ +vvv+
   |   | |   |
   +---+ +---+

   so a(1) = 2.

   For convenience, we will group the polyominoes without
   particular orientation of the curves, and just indicate
   the number of distinct polyominoes of this type, so a(1) becomes

   +***+ (2)
   |   |
   +---+

                 +-+-+         +-+
                 * | *         | *
   +***+***+ (3) * | * (3) +***+ * (4)
   |   |   |     * | *     |   | *
   +---+---+     +-+-+     +---+-+

   so a(2) = 10.

                             +-+         +***+     +-+   +-+
                             | *         |   |     * |   | *
   +***+***+***+ (6) +***+***+ * (8) +---+---+ (8) * +***+ * (6)
   |   |   |   |     |   |   | *     |   |   |     * |   | *
   +---+---+---+     +---+---+-+     +***+***+     |-+---+-+

         +-+         +-+         +***+      +-+           +-+---+
         * |         | *         |   |      * |           * |   |
         * |         | *       +-+-+-+      * +***+       * +***+
         * |         | *       * | *        * |   |       * |
   +---+-+-+ (8) +---+-+-+ (8) * | *  (8) +-+-+---+ (8) +-+-+    (8)
   |   |   |     |   |   |     * | *      |   |         |   |
   +***+***+     +***+***+     +-+-+      +***+         +***+
                                          
       +-+         +-+        
       * |         * |        
       * |         * +---+    
       * |         * |   |    
   +---+-+-+ (8) +-+-+***+ (8)
   |   |   |     |   |        
   +***+***+     +***+        

   so a(3) = 84.  The last two patterns, or 16 polyominos, are
   the ones that are not included in A056755.