A000577 -id:A000577 - cp's OEIS frontend

A137228 Minimal total number of edges in a polyiamond consisting of n triangular cells.

3, 5, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 38, 39, 41, 42, 44, 46, 47, 49, 51, 52, 54, 55, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73, 75, 76, 78, 79, 81, 83, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103, 105, 106, 108, 109, 111, 113

Offset: 1

Tanya Khovanova, Mar 07 2008

nonn

Cf. A135708, A067628, A000577.

a(n) = (3n + A067628(n))/2.

A239658 Number of free triangular n-polyplets (triangular polyominoes connected at edges or corners), where turning over is allowed and holes are allowed.

Original entry on oeis.org

1, 1, 3, 11, 75, 528, 4584, 40609, 373981, 3493723, 33079702, 316109657, 3044648053

Offset: 0

Abe Wits and Ragnar Groot Koerkamp, Nov 20 2020

We get triangular polyominoes when connecting over edges only (rather than edges and corners); that is A000577. The square equivalent (that is, n-polyplets) is A030222. The neighborhood is also known as the Triangular Moore neighborhood, and equals the moves of the Tri-Chess king.

Cf. A000577, A030222.

a(11) and a(12) from Aaron N. Siegel, May 22 2022

A255894 Polyiamond Family Planners: a(n) is the least number of children of a polyiamond of size n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 2, 2, 2, 5, 1, 3

Offset: 0

Gordon Hamilton, Mar 09 2015

			a(7) = 3 because the 7-triangle polyiamond
  ____________
  \          /\
   \        /  \
    \      /    \  a
     \    /      \
      \  /        \      b
       \/__________\___________
       /\          /\          /
      /  \        /  \        /
     /    \      /    \      /
    /      \    /      \    /  c
   /        \  /        \  /
  /__________\/__________\/
  \          /
   \        /
    \      /
     \    /
      \  /
       \/
has three 8-triangle children created by adding a triangle to position a, b or c.

Cf. A000577, A255890.

A284385 Number of fixed convex n-iamonds (or polyiamonds with n cells).

Original entry on oeis.org

2, 3, 6, 8, 6, 7, 12, 15, 8, 9, 18, 18, 8, 15, 24, 23, 12, 12, 24, 30, 12, 17, 36, 31, 14, 21, 30, 30, 18, 24, 42, 42, 14, 21, 48, 38, 14, 33, 48, 45, 24, 18, 42, 54, 24, 35, 66, 48, 20, 36, 48, 44, 24, 40, 72, 63, 18, 27, 72, 60, 26, 51, 66, 62, 36, 30, 60

Offset: 1

Rémy Sigrist, Mar 26 2017

nonn

Also, the number of ways to write n as p^2-q^2-r^2-s^2 with 0 <= min(q, r, s) and max(q, r, s) < n and max(q+r, r+s, s+q) <= n.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, C program for A284385
Eric Weisstein's World of Mathematics, Polyiamond
Wikipedia, Polyiamond
Indranil Ghosh, Python Program to compute the first 1000 terms (translated from the C program by _Rémy Sigrist_)

Cf. A000577, A001420, A096004.

A333018 Number of free heptagonal polyforms with n cells on the heptagonal tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 3, 10, 44, 249, 1513, 9992, 68305, 480748, 3450793, 25186583

Offset: 0

Peter Kagey, Mar 05 2020

The heptagonal tiling is represented by Schläfli symbol {7,3}.
This sequence is to A259352 what A000228 is to A108070.
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, Heptagonal tiling
Wikipedia, Polyform

Analogs with different Schläfli symbols are A000105 ({4,4}), A000207 ({3,oo}), A000228 ({6,3}), A000577 ({3,6}), A005036 ({4,oo}), A119611 ({4,5}), A330659 ({3,7}), and A332930 ({4,6}).

Cf. A108070, A259352.

a(9)-a(12) from Ed Wynn, Feb 14 2021

A337415 a(n) is the number of ways to tile a size n triangle in the triangular grid with n n-cell polyiamonds, up to rotation and reflection of the triangle.

Original entry on oeis.org

1, 0, 1, 3, 11, 314

Offset: 1

Peter Kagey and Alec Jones, Aug 26 2020

The initial values are computed via the program from Arnauld Chevallier in the Code Golf Stack Exchange link.

			An example of the a(4) = 3 essentially different ways of tiling a size 4 triangle with 4-iamonds can be found via the Code Golf Stack Exchange link.

Code Golf Stack Exchange, Cut a triangle into equal-sized parts!
Peter Kagey, Example of the a(5) = 11 partitions of the size 5 triangle into five 5-iamonds.

A328020 is the analog for square on the square grid.

Cf. A000577.

A345076 Number of generalized polyforms on the elongated triangular tiling with n cells.

Original entry on oeis.org

1, 2, 3, 5, 13, 32, 96, 283, 907, 2929, 9787, 32939, 112476, 386230, 1336150, 4642930, 16208851, 56786242, 199614651, 703678568, 2487109359, 8811020024, 31281360326, 111272475650

Offset: 0

Drake Thomas, Jun 07 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.

			See the PDF in the links section.

Peter Kagey, The a(3) = 5 generalized polyforms on the elongated triangular tiling with 3 cells.
Wikipedia, Elongated Triangular 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).

a(15)-a(23) from Bert Dobbelaere, Jun 05 2025

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).

A359525 Number of well-connected free polyiamonds with n cells, where a polyiamond is defined as well-connected if, for all the triangles it contains, the removal of that single triangle would result in an intact polyiamond.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 3, 1, 3, 2, 14

Offset: 1

John Mason, Jan 05 2023

John Mason, Well-connected polyiamond examples

Cf. A000577, A341219.

cp's OEIS Frontend

A137228 Minimal total number of edges in a polyiamond consisting of n triangular cells.

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A239658 Number of free triangular n-polyplets (triangular polyominoes connected at edges or corners), where turning over is allowed and holes are allowed.

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A255894 Polyiamond Family Planners: a(n) is the least number of children of a polyiamond of size n.

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Examples

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A284385 Number of fixed convex n-iamonds (or polyiamonds with n cells).

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Examples

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A333018 Number of free heptagonal polyforms with n cells on the heptagonal tiling of the hyperbolic plane.

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Extensions

A337415 a(n) is the number of ways to tile a size n triangle in the triangular grid with n n-cell polyiamonds, up to rotation and reflection of the triangle.

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Author

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Examples

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Crossrefs

A345076 Number of generalized polyforms on the elongated triangular tiling with n cells.

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Author

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Comments

Examples

Links

Crossrefs

Extensions

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

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Keywords

Crossrefs

Formula

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

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Formula

A359525 Number of well-connected free polyiamonds with n cells, where a polyiamond is defined as well-connected if, for all the triangles it contains, the removal of that single triangle would result in an intact polyiamond.

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