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.

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A197462 Number of free poly-[4.6.12]-tiles (holes allowed) with n cells (division into triangles is significant).

Original entry on oeis.org

1, 3, 3, 9, 14, 38, 74, 185, 414, 1026, 2440, 6077, 14926, 37454, 93749, 237035, 599815, 1526020, 3889117, 9944523, 25475398, 65416733, 168277945, 433705325, 1119610147, 2894928713
Offset: 1

Views

Author

Joseph Myers, Oct 15 2011

Keywords

Comments

[4.6.12] refers to the face configuration of the kisrhombille tiling. - Peter Kagey, May 10 2021

References

  • Branko Gruenbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Links

Crossrefs

Cf. A197463, A197464.
Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Extensions

a(20)-a(26) from Aaron N. Siegel, Jun 03 2022

A197156 Number of free poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.

Original entry on oeis.org

1, 3, 5, 20, 56, 225, 819, 3333, 13336, 55231, 229146, 963284, 4068503, 17301000, 73893082, 317013121, 1364917667, 5896350458, 25545737979, 110968732581
Offset: 1

Views

Author

Joseph Myers, Oct 10 2011

Keywords

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Links

Crossrefs

Cf. A197157, A197158.
Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Extensions

a(16)-a(20) from Bert Dobbelaere, Jun 02 2025

A309159 Number of generalized polyforms on the snub square tiling with n cells.

Original entry on oeis.org

1, 2, 2, 4, 10, 28, 79, 235, 720, 2254, 7146, 22927, 74137, 241461, 790838, 2603210, 8604861, 28549166, 95027832, 317229779, 1061764660, 3562113987, 11976146355
Offset: 0

Views

Author

Peter Kagey and Peter Taylor, Jul 15 2019

Keywords

Comments

The generalized polyforms counted by this sequence are "free", which means that they are counted up to rotation and reflection.

Links

Crossrefs

Cf. A000105, A000228, A000577.

Extensions

a(19)-a(22) from Christian Sievers

A343398 Number of generalized polyforms on the trihexagonal tiling with n cells.

Original entry on oeis.org

1, 2, 1, 4, 9, 30, 97, 373, 1405, 5630, 22672, 93045, 384403, 1602156, 6712128, 28268504, 119537113, 507375130, 2160476897, 9226446455, 39504435891
Offset: 0

Views

Author

Peter Kagey, Apr 13 2021

Keywords

Comments

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.

Links

Crossrefs

Same but distinguishing mirror images: A350739.
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), A343406 (truncated hexagonal), A343577 (truncated square).

Extensions

a(12)-a(15) from John Mason, Mar 04 2022
a(16)-a(20) from Bert Dobbelaere, Jun 06 2025

A343406 Number of generalized polyforms on the truncated hexagonal tiling with n cells.

Original entry on oeis.org

1, 2, 2, 9, 40, 218, 1377, 9285, 65039, 465888, 3385778, 24864272, 184115213, 1372589329, 10291503008, 77544953479
Offset: 0

Views

Author

Peter Kagey, Apr 14 2021

Keywords

Comments

Equivalently, the number of polyhexes with n-k cells and k distinguished vertices.
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.

Links

Crossrefs

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), A343577 (truncated square).

Extensions

a(10)-a(15) from Bert Dobbelaere, Jun 06 2025

A103465 Number of polyominoes that can be formed from n regular unit pentagons (or polypents of order n).

Original entry on oeis.org

1, 1, 2, 7, 25, 118, 551, 2812, 14445, 76092, 403976, 2167116, 11698961, 63544050, 346821209, 1901232614
Offset: 1

Views

Author

Sascha Kurz, Feb 07 2005; definition revised and sequence extended Apr 12 2006 and again Jun 09 2006

Keywords

Comments

Number of 5-polyominoes with n pentagons. A k-polyomino is a non-overlapping union of n regular unit k-gons.
Unlike A051738, these are not anchored polypents but simple polypents. - George Sicherman, Mar 06 2006
Polypents (or 5-polyominoes in Koch and Kurz's terminology) can have holes and this enumeration includes polypents with holes. - George Sicherman, Dec 06 2007

Examples

			a(3)=2 because there are 2 geometrically distinct ways to join 3 regular pentagons edge to edge.

Links

Crossrefs

Cf. A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A120102, A120103, A120104.
Cf. A000105, A000577, A000228.

Extensions

Entry revised by N. J. A. Sloane, Jun 18 2006

A103473 Number of polyominoes consisting of 7 regular unit n-gons.

Original entry on oeis.org

24, 108, 551, 333, 558, 1605, 4418, 8350, 17507, 13512, 17775, 30467, 55264, 83252, 134422, 112514, 135175, 195122, 294091, 397852, 566007, 495773, 568602, 751172, 1031920, 1307384, 1729686, 1557663, 1737915, 2169846, 2808616, 3413064
Offset: 3

Views

Author

Sascha Kurz, Feb 07 2005

Keywords

Examples

			a(3)=24 because there are 24 polyiamonds consisting of 7 triangles and a(4)=108 because there are 108 polyominoes consisting of 7 squares.

Links

Crossrefs

Cf. A103469, A103470, A103471, A103472.
Cf. A000577, A000104, A000228, A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A115071, A120102, A120103, A120104.

Extensions

More terms from Sascha Kurz, Jun 09 2006

A071332 Number of polyiamonds with n cells that tile the plane.

Original entry on oeis.org

1, 1, 1, 3, 4, 12, 23, 66, 139, 341, 567, 2034, 2495, 6354, 12908, 30261, 26556, 110145, 95967, 377523, 499672, 788726, 845130, 4998370, 3694670, 7406217, 13175181, 33557076, 22381719, 117770863
Offset: 1

Views

Author

Joseph Myers, May 19 2002

Keywords

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

Crossrefs

Equals A000577-A071333 and equals A070765-A071334, cf. A054359, A070766.

Extensions

More terms from Joseph Myers, Nov 11 2003
a(29) and a(30) from Joseph Myers, Nov 21 2010

A030223 Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 12, 13, 30, 36, 80, 97, 213, 266, 578, 737, 1589, 2051, 4408, 5747, 12333, 16213, 34737, 45979, 98367, 131007, 279902, 374781, 799732, 1075793, 2293193, 3097415, 6596787, 8942350, 19031088, 25880367, 55043561, 75068945, 159570624, 218189681
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

These are the achiral polyominoes of the regular tiling with Schläfli symbol {3,6}. An achiral polyomino is identical to its reflection. This sequence can most readily be calculated by enumerating achiral fixed polyominoes for three situations with a given axis of symmetry: 1) fixed polyominoes with an axis of symmetry composed of cell edges, A364485; 2) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and a vertex as the highest polyomino point on this axis, A364486; and 3) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and an edge center as the highest polyomino point on this axis, A364487. Those three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Jul 26 2023

Links

Crossrefs

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A001420 (fixed).
Calculation components: A364485, A364486, A364487.
Other tilings: A030227 {4,4}, A030225 {6,3}.

Formula

From Robert A. Russell, Jul 27 2023: (Start)
a(n) = (A364486(n) + A364487(n)) / 2, n odd.
a(n) = (A364485(n/2) + A364486(n) + A364487(n)) / 2, n even.
a(n) = 2*A000577(n) - A006534(n) = A006534(n) - 2*A030224(n) = A000577(n) - A030224(n). (End)

Extensions

a(19) to a(28) from Joseph Myers, Sep 24 2002
Additional terms from Robert A. Russell, Jul 26 2023
Name edited by Robert A. Russell, Jul 27 2023

A067628 Minimal perimeter of polyiamond with n triangles.

Original entry on oeis.org

0, 3, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 14, 15, 16, 15, 16, 15, 16, 17, 16, 17, 16, 17, 18, 17, 18, 17, 18, 19, 18, 19, 18, 19, 18, 19, 20, 19, 20, 19, 20, 21, 20, 21, 20, 21, 20, 21, 22, 21, 22, 21, 22
Offset: 0

Views

Author

Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002

Keywords

Comments

A polyiamond is a shape made up of n congruent equilateral triangles.

References

  • Frank Harary and Heiko Harborth, Extremal animals, J. Combinatorics Information Syst. Sci., 1(1):1-8, 1976.

Links

Crossrefs

Cf. A000105, A000577, A027709 (squares), A057729, A135711.

Programs

  • Maple
    interface(quiet=true); for n from 0 to 100 do if (1 = 1) then temp1 := ceil(sqrt(6*n)); end if; if ((temp1 mod 2) = (n mod 2)) then temp2 := 0; else temp2 := 1; end if; printf("%d,", temp1 + temp2); od;
  • PARI
    a(n)=2*ceil((n+sqrt(6*n))/2)-n; \\ Stefano Spezia, Oct 02 2019
  • Python
    from math import isqrt
def A067628(n): return (c:=isqrt(6*n-1)+1)+((c^n)&1) if n else 0 # Chai Wah Wu, Jul 28 2022

Formula

Let c(n) = ceiling(sqrt(6n)). Then a(n) is whichever of c(n) or c(n) + 1 has the same parity as n.
a(n) = 2*ceiling((n + sqrt(6*n))/2) - n (Harary and Harborth, 1976). - Stefano Spezia, Oct 02 2019
Previous Showing 11-20 of 70 results. Next

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