Polycubes: (upto 4 cubes)

  LEX = Lexicographic Number of the polycube

  LEX   Coordinates          Picture

  1
  1-1   000                  1

  2
  1-2   000 001              1 1

  3
  1-3   000 001 002          1 1 1

  4                          1
  2-3   000 001 010          1 1

  5
  1-4   000 001 002 003      1 1 1 1

  6                          1
  2-4   000 001 002 010      1 1 1

  7                            1
  3-4   000 001 002 011      1 1 1

  8                          1 1
  4-4   000 001 010 011      1 1

  9                          1
  5-4   000 001 010 100      2 1

 10                          1         -10       1
  6-4   000 001 010 101      1 2        -6-4   2 1

 11                            1 1
  7-4   000 001 011 012      1 1


  n    Polycubes/O_3     Polycubes/O_3+
          A038119           A000162

  1          1                 1
  2          1                 1
  3          2                 2
  4          7                 8
  5         23                29
  6        112               166
  7        607              1023
  8       3811              6922
  9      25413             48311
 10     178083            346543
 11    1279537           2522522
 12    9371094          18598427

References:

- W. F. Lunnon;
  Symmetry of Cubical and General Polyominoes,
  In: Ronald C. Read; Graph Theory and Computing,
  Academic Press, New York, 1972, 101-108
  [counts Polycubes with n<=6]

- W. F. Lunnon;
  Counting Multidimensional Polyominoes, 
  Computer Journal 18 (1975), no. 4, 366-367. 
--
mailto:Torsten.Sillke@uni-bielefeld.de
http://www.mathematik.uni-bielefeld.de/~sillke/