A000104 Number of n-celled free polyominoes without holes.

1, 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4460, 16094, 58937, 217117, 805475, 3001127, 11230003, 42161529, 158781106, 599563893, 2269506062, 8609442688, 32725637373, 124621833354, 475368834568, 1816103345752, 6948228104703, 26618671505989, 102102788362303

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

References

• J. S. Madachy, Pentominoes - Some Solved and Unsolved Problems, J. Rec. Math., 2 (1969), 181-188.
• George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• John Mason, Table of n, a(n) for n = 0..40
• Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871.
• John Mason, Description of counting programs
• John Mason, Programs for calculation of numbers of unholey polyominoes
• Lucia Moura and Ivan Stojmenovic, Backtracking and Isomorph-Free Generation of Polyhexes, Table 2.2 on p. 55 of Handbook of Applied Algorithms (2008).
• W. R. Muller, K. Szymanski, J. V. Knop, and N. Trinajstic, On the number of square-cell configurations, Theor. Chim. Acta 86 (1993) 269-278
• Tomás Oliveira e Silva, Enumeration of polyominoes
• T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy)
• R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Crossrefs

Cf. A000105, row sums of A308300, A006746, A056877, A006748, A056878, A006747, A006749, A054361, A070765 (polyiamonds), A018190 (polyhexes), A266549 (by perimeter).

Cf. A056879, A056880, A056881, A056882, A006724, A357647, A357648.

Formula

a(n) = A000105(n) - A001419(n). - John Mason, Sep 06 2022

a(n) = (4*A056879(n) + 4*A056881(n) + 4*A056883(n) + 6*A056880(n) + 6*A056882(n) + 6*A357647(n) + 7*A357648(n) + A006724(n)) / 8. - John Mason, Oct 10 2022

Extensions

Extended to n=26 by Tomás Oliveira e Silva

a(27)-a(28) from Tomás Oliveira e Silva's page added by Andrey Zabolotskiy, Oct 02 2022