cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 61-70 of 70 results.

A364684 Number of achiral triangular polyominoes with 6n cells and sixfold rotational symmetry.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 7, 9, 16, 22, 46, 63, 121, 167, 455, 912, 1263, 2535, 3514, 7099, 9873, 20043, 27956, 56807, 79397, 161736, 226559, 462482, 649100, 1327165, 1865833, 3820605, 5379507, 11028753, 15550459, 31913892, 45057416, 92557088, 130837407, 268988726
Offset: 1

Views

Author

Robert A. Russell, Aug 02 2023

Keywords

Comments

These are polyominoes of the regular tiling with Schläfli symbol {3,6}. Their center is a vertex. As their symmetry group of order 12 is maximal, we can consider them either fixed, oriented (one-sided), or unoriented (free).

Examples

			                            ________          ________
                /\         /\  /\  /\        /\  /\  /\
    ____   ____/__\____   /__\/__\/__\      /__\/__\/__\
   /\  /\  \  /\  /\  /  /\  /    \  /\    /\  /\  /\  /\
  /__\/__\  \/__\/__\/  /__\/      \/__\  /__\/__\/__\/__\
  \  /\  /  /\  /\  /\  \  /\      /\  /  \  /\  /\  /\  /
   \/__\/  /__\/__\/__\  \/__\____/__\/    \/__\/__\/__\/
               \  /       \  /\  /\  /      \  /\  /\  /
                \/         \/__\/__\/        \/__\/__\/

Links

Crossrefs

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A030223 (achiral), A001420 (fixed).

A057782 Building block is trapezoid formed from 3 equilateral triangles; sequence gives number of pieces (polytraps) that can be formed from n such trapezoids.

Original entry on oeis.org

1, 9, 94, 1552, 27285, 509805, 9783124
Offset: 1

Views

Author

N. J. A. Sloane, Oct 29 2000

Keywords

Comments

Also known as "Polytriamonds" because the building block is the unique triamond (composite of three equilateral triangles joined edge-to-edge). - Aaron N. Siegel, May 23 2022

References

  • Computed by Brendan Owen.

Links

Crossrefs

Cf. A000577, A000228, A000105, A056844.

Extensions

Link updated by William Rex Marshall, Dec 16 2009
a(7) from Aaron N. Siegel, May 23 2022

A103464 Number of polyominoes consisting of n regular unit heptagons.

Original entry on oeis.org

1, 1, 2, 7, 25, 118, 558, 2876, 14982, 80075, 431889, 2354991, 12930257, 71459124, 396978189, 2215609864
Offset: 3

Views

Author

Sascha Kurz, Jun 09 2006

Keywords

Links

Crossrefs

Cf. A000577, A000104, A000228, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A120102, A120103, A120104.

A119532 Number of n-ominoes plus number of n-iamonds.

Original entry on oeis.org

2, 2, 2, 3, 8, 16, 47, 132, 435, 1445, 5103, 18259, 66934, 247826, 928137, 3500559, 13290552, 50712016, 194358380, 747624825, 2885120934, 11164896416, 43313276948, 168400053019, 656028153120, 2560227845391, 10007858038797, 39178672900385, 153586295761270, 602842558581537, 2368994981198048
Offset: 0

Views

Author

Jonathan Vos Post, May 28 2006

Keywords

Comments

Number of polyominoes with n cells plus number of polyiamonds with n cells. Turning over is allowed; holes are allowed.

Crossrefs

Cf. A000105, A000577.

Formula

a(n) = A000105(n) + A000577(n).

Extensions

More terms from John Mason, Jan 08 2023

A131486 a(n) is the number of triangular polyominoes (polyiamonds) with n edges.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 4, 1, 11, 1, 23, 4, 62, 11, 150, 38, 411, 118, 1081, 389
Offset: 1

Views

Author

Tanya Khovanova, Jul 28 2007

Keywords

Comments

An n-celled polyiamond with perimeter p has (3n+p)/2 edges. The maximum number of edges in an n-celled polyiamond is 2n+1.

Links

Crossrefs

Cf. A000577: Triangular polyominoes (or polyiamonds) with n cells. A057729: Number of triangular polyominoes (or polyiamonds) [A000577] with perimeter n. A131481: a(n) is the number of n-cell polyiamonds (triangular polyominoes) with perimeter n+2.

A137191 Number of n-celled polyiamonds with perimeter < n+2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 11, 39, 120, 403, 1254, 4000, 12475, 39030
Offset: 1

Views

Author

Tanya Khovanova, Mar 03 2008

Keywords

Comments

These are n-cell polyiamonds with the perimeter less than the maximum possible one.
If we associate a graph to a polyiamond with vertices representing cells and edges representing two cells with a common edge, then this sequence enumerates polyiamonds whose corresponding graphs have cycles.

Formula

A000577(n) - A131481(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

Views

Author

Stefano Spezia, Sep 29 2019

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Formula

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

A330211 Number of free pentagonal polyforms with n cells on the order-4 pentagonal tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 2, 8, 28, 143, 747, 4346, 25974, 160869, 1015723, 6531611, 42592880
Offset: 0

Views

Author

Peter Kagey, Mar 05 2020

Keywords

Comments

The order-4 pentagonal tiling of the hyperbolic plane has Schläfli symbol {5,4}.
This sequence is computed from via program by Christian Sievers in the Code Golf Stack Exchange link.

Links

Crossrefs

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}), A332930 ({4,6}), and A333018 ({7,3}).

Programs

  • C
    // See the Code Golf link.
  • GAP
    # See the Code Golf link.
  • bc
    # See the Code Golf link.

Extensions

a(8)-a(13) from Ed Wynn, Feb 16 2021

A359526 Number of unbiased (balanced) free polyiamonds with 2n cells.

Original entry on oeis.org

1, 2, 10, 44, 283, 1922, 14163, 107771, 841846
Offset: 1

Views

Author

John Mason, Jan 05 2023

Keywords

Comments

Define the bias of a polyiamond to be the difference between the number of black triangles and the number of white triangles when chessboard coloring is applied to the polyiamond. Unbiased polyiamonds are those having an equal number of black and white triangles. The sequence is defined for 2n rather than n as odd-sized polyiamonds are obviously biased.

Crossrefs

Cf. A000577, A234012.

A359688 a(n) is the number of asymmetrical polyiamonds of n cells.

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 36, 94, 294, 794, 2300, 6394, 18266, 51592, 147426, 420512, 1206740, 3466876
Offset: 1

Views

Author

John Mason, Jan 11 2023

Keywords

Crossrefs

Cf. A000577.
Previous Showing 61-70 of 70 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.