A000577 -id:A000577 - cp's OEIS frontend

A069813 Maximum number of triangles in polyiamond with perimeter n.

1, 2, 3, 6, 7, 10, 13, 16, 19, 24, 27, 32, 37, 42, 47, 54, 59, 66, 73, 80, 87, 96, 103, 112, 121, 130, 139, 150, 159, 170, 181, 192, 203, 216, 227, 240, 253, 266, 279, 294, 307, 322, 337, 352, 367, 384, 399, 416, 433, 450, 467, 486, 503, 522, 541, 560, 579

Offset: 3

Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

			a(10) = 16 because the maximum number of triangles in a polyiamond of perimeter 10 is 16.

Colin Barker, Table of n, a(n) for n = 3..1000
W. C. Yang, R. R. Meyer, Maximal and minimal polyiamonds, 2002.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A000105, A000577, A027709, A030511 (bisection), A057729, A067628.

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)))); // Marius A. Burtea, Jan 03 2020

A069813 := proc(n)
    round(n^2/6) ;
    if modp(n,6) <> 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, Jul 14 2015

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 2, 3, 6, 7, 10}, 60] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = round(n^2/6) - (n % 6 != 0) \\ Michel Marcus, Jul 17 2013

Vec(x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^60)) \\ Colin Barker, Jan 19 2015

a(n) = round(n^2/6) - (0 if n = 0 mod 6, 1 else) = A056829(n)-A097325(n).
From Colin Barker, Jan 18 2015: (Start)
a(n) = round((-25 + 9*(-1)^n + 8*exp(-2/3*i*n*Pi) + 8*exp((2*i*n*Pi)/3) + 6*n^2)/36), where i=sqrt(-1).
G.f.: x^3*(1+x-x^2)*(1+x^2) / ((1-x)^3*(1+x)*(1+x+x^2)). (End)
a(n) = A001399(n-3) + A001399(n-4) + A001399(n-6) - A001399(n-7). - R. J. Mathar, Jul 14 2015

A344211 Number of generalized polyforms on the rhombitrihexagonal tiling with n cells.

Original entry on oeis.org

1, 3, 2, 7, 16, 60, 201, 838, 3407, 14767, 64200, 284676, 1269981, 5715325, 25854146, 117576949, 536918541, 2461230475, 11319321354, 52212782646

Offset: 0

Peter Kagey, May 11 2021

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

Kadon Enterprises, Birds and Bees.
Peter Kagey, Illustration of a(3) = 7.
George Sicherman, Catalogue of Polybirds.
Wikipedia, Rhombitrihexagonal tiling
Livio Zucca, PolyTriForms.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(11)-a(19) from Bert Dobbelaere, Jun 05 2025

A369366 Least number of inequivalent cells in a polyiamond with n cells.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 3, 3, 3, 3, 6, 2, 4, 4, 4, 4, 9, 2

Offset: 1

Pontus von Brömssen, Jan 22 2024

Equivalently, a(n) is the least number of free rooted (or pointed) polyiamonds (A369365) corresponding to a given polyiamond with n cells.

Index entries for sequences related to polyominoes.

Cf. A000577, A367758, A369363, A369365, A369367.

a(18) from John Mason, Sep 20 2024

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

A096004 Number of convex triangular polyominoes containing n cells.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 5, 2, 3, 3, 4, 2, 4, 4, 6, 3, 3, 4, 5, 2, 5, 5, 7, 3, 4, 5, 6, 3, 5, 5, 8, 3, 4, 5, 6, 4, 7, 7, 9, 4, 5, 5, 7, 3, 7, 8, 9, 3, 5, 7, 8, 4, 8, 8, 11, 4, 5, 7, 8, 4, 9, 9, 11, 5, 5, 8, 9, 4, 9, 9, 13, 5, 7, 9, 8, 5, 8, 9, 12

Offset: 1

Paul Boddington, Jul 27 2004

The main sequence on triangular polyominoes is A000577. The convexity condition makes enumeration easy as a convex triangular polyomino has at most 6 sides. It is simple to prove that a(n) is also the number of 4-tuples (p,b,c,d) of nonnegative integers satisfying b<=c<=d, b+c+d<=p, n=p^2-b^2-c^2-d^2.

For n = A014529(k) there are a(n) many polygons. At least one of them can be tiled with k equilateral triangles. - Rainer Rosenthal, Sep 20 2017

			a(8)=3 because there are 3 ways to compose a convex polygon of 8 equilateral triangles with side 1:
.
    *---*---*---*---*
   / \ / \ / \ / \ /
  *---*---*---*---*
      *---*---*
     / \ / \ /
    *---*---*
   / \ / \ /
  *---*---*
      *---*
     / \ / \
    *---*---*
   / \ / \ / \
  *---*---*---*

Rainer Rosenthal, Table of n, a(n) for n = 1..5200
Peter Kagey, Examples for a(1)-a(30).
Walter Trump, Number of Convex Polygons of a Given Perimeter or Area on the Triangular Lattice, 2025.

Cf. A000577, A014529.

a:=proc(n) local x,p,d,c,b; x:=0; for p from 0 to ceil((n+1)/2) do; for d from 0 to p do; for c from 0 to min(d,p-d) do; for b from 0 to min(c,p-c-d) do; if p^2-b^2-c^2-d^2=n then x:=x+1 fi; od; od; od; od; x; end; # corrected by Rainer Rosenthal, Sep 20 2017

a(n) >= sqrt(n)/3. - Baohua Tian, Apr 21 2020

a(83) and a(84) corrected by Rainer Rosenthal, Sep 20 2017

A344213 Number of generalized polyforms on the truncated trihexagonal tiling with n cells.

Original entry on oeis.org

1, 3, 3, 14, 50, 261, 1397, 8364, 50643, 315512, 1984853, 12619579, 80802300, 520724842, 3373646060, 21959949924, 143527242317

Offset: 0

Peter Kagey, May 11 2021

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

Peter Kagey, Illustration of a(3) = 14.
Wikipedia, Truncated trihexagonal tiling

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square), A344211 (rhombitrihexagonal).

a(10)-a(16) from Bert Dobbelaere, Jun 07 2025

A383908 Number of generalized polyforms with n cells on the snub trihexagonal tiling.

Original entry on oeis.org

1, 3, 3, 7, 23, 69, 228, 766, 2642, 9309, 33382, 120629, 439752, 1613135, 5953061, 22075011, 82204128, 307213215, 1151820825, 4330858682, 16326297768, 61690058385

Offset: 0

Peter Kagey, May 14 2025

A generalized polyform on the snub trihexagonal tiling with n-cells is a collection of n faces connected edgewise. Two polyforms are considered the same they are related by an isometry (translation and/or rotation) of the snub trihexagonal tiling.

			For n=1, the a(1) = 3 generalized polyforms are the three types of faces: hexagons, hexagon-adjacent triangles, and hexagon-nonadjacent triangles.
For n=2, the a(2) = 3 generalized polyforms are
(1) a hexagon with a hexagon-adjacent triangle,
(2) a hexagon-adjacent triangle with a hexagon-nonadjacent triangle, and
(3) two hexagon-adjacent triangles.

Peter Kagey, Illustration of a(1)-a(4).
Wikipedia, Snub trihexagonal tiling

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square), A344211 (rhombitrihexagonal), A344213 (truncated trihexagonal).

a(12)-a(21) from Bert Dobbelaere, Jun 05 2025

A131481 a(n) is the number of n-cell polyiamonds (triangular polyominoes) with perimeter n+2.

Original entry on oeis.org

1, 1, 1, 3, 4, 11, 23, 62, 149, 409, 1066, 2931, 7981, 22166, 61508, 172267, 483088, 1361475, 3845139, 10894630

Offset: 1

Tanya Khovanova, Jul 27 2007

n+2 is the maximal perimeter for an n-celled polyiamond. a(n) is the number of n-celled polyiamonds that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge)

Andrew Clarke, Isoperimetrical Polyiamonds

a(n) <= A000577(n), a(n) <= A057729(n+2).

Offset corrected and terms a(17)-a(20) added by John Mason, May 15 2021

A330659 The number of free polyiamonds with n cells on an order-7 triangular tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 12, 27, 78, 208, 637, 1870, 5797, 17866, 56237, 177573, 566904, 1818527, 5874180, 19065038

Offset: 0

Peter Kagey, Mar 01 2020

This gives the number of polyforms with n cells in the hyperbolic tiling with Schläfli symbol {3,7}.

This sequence is computed from via program by Christian Sievers in the Code Golf Stack Exchange link.

Code Golf Stack Exchange, Impress Donald Knuth by counting polyominoes on the hyperbolic plane
Wikipedia, Order-7 triangular tiling
Wikipedia, Polyiamond

Analogs with different Schläfli symbols are A000207 ({3,oo}), A000577 ({3,6}), A005036 ({4,oo}), and A119611 ({4,5}).

a(11)-a(19) from Ed Wynn, Feb 14 2021

A367172 a(n) = the maximum number of distinct tilings of a polyiamond of size n using any combination of polyiamond tiles of sizes 1 through n.

Original entry on oeis.org

1, 2, 4, 8, 16, 58, 116, 232, 464, 1690, 3380, 6760, 24712, 49424, 98848, 361258, 722516, 1445032, 5280576, 10561152, 21122304, 77188216

Offset: 1

Craig Knecht and John Mason, Nov 07 2023

The sequence considers reflections and rotations as distinct tilings.

The polyiamonds being tiled and the tiles themselves may be with or without holes.

			a(3) = 4 because the triamond can be tiled by (1) a triamond, (2) 3 moniamonds, and (3) a moniamond and a diamond in two distinct ways.

Craig Knecht, John Mason and Walter Trump, Polyiamond tiling V2

Cf. A000577, A367173, A367174.

More terms from Walter Trump, May 01 2024

cp's OEIS Frontend

A069813 Maximum number of triangles in polyiamond with perimeter n.

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A344211 Number of generalized polyforms on the rhombitrihexagonal tiling with n cells.

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A369366 Least number of inequivalent cells in a polyiamond with n cells.

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

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A096004 Number of convex triangular polyominoes containing n cells.

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A344213 Number of generalized polyforms on the truncated trihexagonal tiling with n cells.

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A383908 Number of generalized polyforms with n cells on the snub trihexagonal tiling.

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A131481 a(n) is the number of n-cell polyiamonds (triangular polyominoes) with perimeter n+2.

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A330659 The number of free polyiamonds with n cells on an order-7 triangular tiling of the hyperbolic plane.