A054361 -id:A054361 - cp's OEIS frontend

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

N. J. A. Sloane

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).

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.

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.

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

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

A054360 Number of polyominoes with n cells that do not tile the plane.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 26, 235, 1585, 10453, 40151, 199922, 792867, 3146993, 12532362, 49280874, 189564720, 739243108, 2858545267, 11101139486, 43154849550, 167993679964, 654778791594, 2556955750961

Offset: 1

Joe Keane (jgk(AT)jgk.org)

			a(7) = 4.

Posting by Glenn C. Rhoads (rhoads(AT)paul.rutgers.edu) to rec.puzzles, Feb 17 2000.

Joseph Myers, Polyomino tiling

Cf. A000105, A054359, A054361.

Corrected and extended by Joseph Myers, May 05 2002
More terms from Joseph Myers, Nov 04 2003
a(24) and a(25) from Joseph Myers, Nov 17 2010

A070768 Number of polyhexes with n cells without holes that do not tile the plane.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 37, 381, 2717, 18760, 116439, 565943, 3033697, 14835067, 72633658, 356923880, 1746833634, 8532601529, 41868336466, 205618704167, 1012359995953

Offset: 1

Joseph Myers, May 05 2002

M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.

Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms
Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms, arXiv:2105.09438 [cs.CG], 2021. See Table 3 and Table 4.
Joseph Myers, Polyhex tiling

Cf. A018190, A070766, A070767, A038144, A054361, A071334.

More terms from Joseph Myers, Nov 06 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

A071334 Number of polyiamonds with n cells without holes that do not tile the plane.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 20, 103, 594, 1192, 6290, 18099, 54808, 159048, 502366, 1374593, 4076218, 11378831, 32674779, 93006494, 264720498, 748062099, 2134512296, 6071524897, 17289205132, 49268564671, 140605019208, 401392287316

Offset: 1

Joseph Myers, May 19 2002

From Bernard Schott, Feb 21 2020: (Start)
There exist 112 polyiamonds without holes that have from 1 to 8 cells (A070765), but only one of these polyiamonds, corresponding to a(7)= 1 cannot tile the plane. This polyiamond is called V-shaped heptiamond (see proof in Martin Gardner's link in German).
  __    __
  \  /\  /\  /
   \/\/\/
    \  /\  /
     \/\/
(End)

M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.

Martin Gardner, V-heptiamond, Mathematisches Labyrinth: Neue Probleme für die Knobelgemeinde, p. 118, Google books.
Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms
Craig S. Kaplan, Heesch Numbers of Unmarked Polyforms, arXiv:2105.09438 [cs.CG], 2021. See Table 5 and Table 6.
Joseph Myers, Polyiamond tiling

Equals A070765-A071332 and A071333-A070764, cf. A054361, A070768.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000577 = A@000577;
A070764 = A@070764;
A071332 = A@071332;
a[n_] := A000577[[n]] - A070764[[n]] - A071332[[n]];
a /@ Range[30] (* Jean-François Alcover, Feb 21 2020 *)

More terms from Joseph Myers, Nov 11 2003

a(29) and a(30) from Joseph Myers, Nov 21 2010

cp's OEIS Frontend

A000104 Number of n-celled free polyominoes without holes.

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A054360 Number of polyominoes with n cells that do not tile the plane.

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A070768 Number of polyhexes with n cells without holes that do not tile the plane.

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A071334 Number of polyiamonds with n cells without holes that do not tile the plane.

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