A070764 -id:A070764 - cp's OEIS frontend

A070765 Number of polyiamonds with n cells, without holes.

1, 1, 1, 3, 4, 12, 24, 66, 159, 444, 1161, 3226, 8785, 24453, 67716, 189309, 528922, 1484738, 4172185, 11756354, 33174451, 93795220, 265565628, 753060469, 2138206966, 6078931114, 17302380313, 49302121747, 140627400927, 401510058179

Offset: 1

Joseph Myers, May 05 2002

If holes are allowed, we get A000577.

Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Riddles of the sphinx tilings, arXiv:2304.14388 [cond-mat.stat-mech], 2023.
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.
Hai Pham-Van, Linh Tran-Phan-Thuy, Cuong Tran-Manh, Bich Do-Danh, and Hoang Luc-Huy, Two-Dimensional Clusters of Colloidal Particles Induced by Emulsion Droplet Evaporation, Nanomaterials (2020) Vol. 10, 156.

Equals A000577(n)-A070764(n). Cf. A000104, A018190.

More terms from Joseph Myers, Nov 11 2003

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

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

A071333 Number of polyiamonds with n cells that do not tile the plane.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 21, 107, 619, 1300, 6740, 19812, 61075, 181036, 577551, 1626183, 4904626, 14071461, 41336066, 120630534, 352200161, 1023454347, 2997105957, 8761810505, 25648786717, 75161609591, 220583137840, 647825855933

Offset: 1

Joseph Myers, May 19 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.

Joseph Myers, Polyiamond tiling

Equals A000577(n)-A071332(n) and A070764(n)+A071334(n). Cf. A054360, A070767.

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;
A071332 = A@071332;
a[n_] := A000577[[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

A359523 Number of free polyiamonds of n cells with multiply-connected interiors.

Original entry on oeis.org

1, 2, 12, 51, 217, 846, 3211

Offset: 12

John Mason, Jan 05 2023

Cf. A000577, A070764, A359519, A359524.

a(n) = A070764(n) - A359524(n).

A359524 Number of free holey polyiamonds of n cells with simply-connected interiors.

Original entry on oeis.org

1, 4, 25, 107, 448, 1701, 6216, 21771, 74339, 248379

Offset: 9

John Mason, Jan 05 2023

Cf. A000577, A070764, A359520, A359523.

a(n) = A070764(n) - A359523(n).

A327896 a(n) is the minimum number of tiles needed for constructing a polyiamond with n holes.

Original entry on oeis.org

9, 14, 19, 23, 27, 31, 35, 39, 43, 47, 51, 54, 58, 62, 65, 69, 73, 76, 80, 83, 87, 90, 94, 97, 101, 104, 108, 111, 115, 118, 122, 125, 129, 132, 135, 139, 142, 146, 149, 152, 156, 159, 163, 166, 169, 173, 176, 179, 183, 186, 189, 193, 196, 199, 203, 206, 209, 213

Offset: 1

Stefano Spezia, Sep 29 2019

nonn

For n > 0, it is easy to prove that k(n) = floor((3 + sqrt(3*(3+8*n)))/6) is the unique integer that satisfies the inequalities 3*binomial(k,2) <= n <= 3*binomial(k+1,2) of Theorem 1.1 in Malen and Roldán.

Proof: solving in k the above inequalities for n > 0, one gets that x - 1 <= k <= x, where x = (3 + sqrt(3*(3+8*n)))/6. Since 3*(3+8*n) is never a perfect square, it follows that x is not an integer and k = floor(x). QED.

Greg Malen and Érika Roldán, Polyiamonds Attaining Extremal Topological Properties, arXiv:1906.08447 [math.CO], 2019.

Cf. A000217, A000577, A067628, A069813, A070764, A161680.

k:=n->floor((3+sqrt(3*(3+8*n)))/6): a:=n->3*(n+k(n))+1+ceil(2*n/k(n)): seq(a(n), n = 1 .. 58)

k[n_]:=Floor[(3+Sqrt[3*(3+8n)])/6]; a[n_]:=3(n+k[n])+1+Ceiling[2n/k[n]]; Array[a,58]

a(n) = 3*(n + k(n)) + 1 + ceiling(2*n/k(n)), where k(n) = floor((3 + sqrt(3*(3+8*n)))/6).

cp's OEIS Frontend

A070765 Number of polyiamonds with n cells, without holes.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A071333 Number of polyiamonds with n cells that do not tile the plane.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Extensions

A359523 Number of free polyiamonds of n cells with multiply-connected interiors.

Views

Author

Keywords

Crossrefs

Formula

A359524 Number of free holey polyiamonds of n cells with simply-connected interiors.

Views

Author

Keywords

Crossrefs

Formula

A327896 a(n) is the minimum number of tiles needed for constructing a polyiamond with n holes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula