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

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

1,9

Comments

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).

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References

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.

Links

Table of n, a(n) for n=1..30.

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A113599 A077757 A126859 * A241355 A220207 A189950

Adjacent sequences: A071331 A071332 A071333 * A071335 A071336 A071337

Keyword

nonn,hard,more

Author

Joseph Myers, May 19 2002

Extensions

More terms from Joseph Myers, Nov 11 2003

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

Status

approved