A001419 - OEIS

A001419

Number of n-celled polyominoes with holes.
(Formerly M4226 N1767)

0, 0, 0, 0, 0, 0, 1, 6, 37, 195, 979, 4663, 21474, 96496, 425449, 1849252, 7946380, 33840946, 143060339, 601165888, 2513617990, 10466220315, 43425174374, 179630865835, 741123699012, 3050860717372, 12534339432498, 51408312232300, 210526591157926, 860989703302456

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

From John Mason, Sep 06 2022: (Start)

Conjecture: Almost all polyominoes are holey. In other words, a(n)/A000105(n) tends to 1 for increasing n.

The number of holes in a polyomino is given by the formula (based on a generalization of Pick's Theorem): H = n + 1 - i - s / 2, where:

n is the size (area) of the polyomino;

i is the number of completely internal vertices; i.e., the number of vertices that are surrounded by 4 squares;

s is the number of vertices on a single border; i.e., vertices that are the corners of 1, 2 or 3 squares, but excluding those that touch only 2 squares that are diagonally adjacent. (End)

REFERENCES

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition ( Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.

Joseph 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 = 1..40

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.

Joseph Myers, Polyomino tiling

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.

Eric Weisstein's World of Mathematics, Polyomino.

Wikipedia, The 6 Octominoes with holes

Wikipedia, The 37 Nonominoes with holes

FORMULA

a(n) >= A057418(n). - R. J. Mathar, Jun 15 2014

a(n) = A000105(n) - A000104(n). - Jean-François Alcover, Jan 04 2020, after R. J. Mathar in A000105.

MATHEMATICA

A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];

A000104 = A@104;

A000105 = A@105;

a[n_] := A000105[[n + 1]] - A000104[[n + 1]];

a /@ Range[40] (* Jean-François Alcover, Jan 04 2020, updated Apr 21 2024 after John Mason's b-file *)

CROSSREFS

Cf. A000104, A000105.

Sequence in context: A056328 A156185 A057418 * A081152 A244618 A033116

Adjacent sequences: A001416 A001417 A001418 * A001420 A001421 A001422

KEYWORD

nonn,hard

AUTHOR

N. J. A. Sloane