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.

Showing 1-9 of 9 results.

A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

Original entry on oeis.org

1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875, 372868101, 1822236628, 8934910362, 43939164263, 216651036012, 1070793308942, 5303855973849, 26323064063884, 130878392115834, 651812979669234, 3251215493161062, 16240020734253127, 81227147768301723, 406770970805865187, 2039375198751047333
Offset: 1

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Author

Keywords

Comments

From Markus Voege, Nov 24 2009: (Start)
On the difference between this sequence and A038147:
The first term that differs is for n=6; for all subsequent terms, the number of polyhexes is larger than the number of planar polyhexes.
If I recall correctly, polyhexes are clusters of regular hexagons that are joined at the edges and are LOCALLY embeddable in the hexagonal lattice.
"Planar polyhexes" are polyhexes that are GLOBALLY embeddable in the honeycomb lattice.
Example: (Planar) polyhex with 6 cells (x) and a hole (O):
.. x x
. x O x
.. x x
Polyhex with 6 cells that is cut open (I):
.. xIx
. x O x
.. x x
This polyhex is not globally embeddable in the honeycomb lattice, since adjacent cells of the lattice must be joined. But it can be embedded locally everywhere. It is a start of a spiral. For n>6 the spiral can be continued so that the cells overlap.
Illegal configuration with cut (I):
.. xIx
. x x x
.. x x
This configuration is NOT a polyhex since the vertex at
.. xIx
... x
is not embeddable in the honeycomb lattice.
One has to keep in mind that these definitions are inspired by chemistry. Hence, potential molecules are often the motivation for these definitions. Think of benzene rings that are fused at a C-C bond.
The (planar) polyhexes are "free" configurations, in contrast to "fixed" configurations as in A001207 = Number of fixed hexagonal polyominoes with n cells.
A000228 (planar polyhexes) and A001207 (fixed hexagonal polyominoes) differ only by the attribute "free" vs. "fixed," that is, whether the different orientations and reflections of an embedding in the lattice are counted.
The configuration
. x x .... x
.. x .... x x
is counted once as free and twice as fixed configurations.
Since most configurations have no symmetry, (A001207 / A000228) -> 12 for n -> infinity. (End)

References

  • A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
  • A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, 3599-3609
  • M. Gardner, Polyhexes and Polyaboloes. Ch. 11 in Mathematical Magic Show. New York: Vintage, pp. 146-159, 1978.
  • 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.
  • J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
  • W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Extensions

a(13) from Achim Flammenkamp, Feb 15 1999
a(14) from Brendan Owen, Dec 31 2001
a(15) from Joseph Myers, May 05 2002
a(16)-a(20) from Joseph Myers, Sep 21 2002
a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(22)-a(30) from John Mason, Jul 18 2023

A197549 Number of free poly-IH10-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 5, 16, 62, 276, 1222, 5563, 25805, 120909, 572011, 2727485, 13089106, 63164265
Offset: 1

Views

Author

Joseph Myers, Oct 16 2011

Keywords

Comments

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a rotation of order 3.

References

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

Crossrefs

A197550 Number of free poly-IH8-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 3, 7, 28, 99, 433, 1852, 8463, 38798, 181889, 858570, 4093739, 19636172, 94759074
Offset: 1

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Author

Joseph Myers, Oct 16 2011

Keywords

Comments

Equivalently, one-sided polybricks, or polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a rotation of order 2.

References

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

Crossrefs

A197551 Number of free poly-IH18-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 10, 37, 147, 637, 2823, 13020, 60649, 286549, 1364667, 6547108, 31586568
Offset: 1

Views

Author

Joseph Myers, Oct 16 2011

Keywords

Comments

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation, a rotation of order 3 or a reflection or glide reflection in a line perpendicular to the sides of the hexagons.

References

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

Crossrefs

A197552 Number of free poly-IH19-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 10, 36, 146, 631, 2815, 12987, 60601, 286376, 1364399, 6546220, 31585133
Offset: 1

Views

Author

Joseph Myers, Oct 16 2011

Keywords

Comments

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation, a rotation of order 3 or a reflection or glide reflection in a line parallel to the sides of the hexagons.

References

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

Crossrefs

A197553 Number of free poly-IH12-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 7, 24, 99, 416, 1852, 8386, 38797, 181530, 858560, 4092031, 19636098, 94750833
Offset: 1

Views

Author

Joseph Myers, Oct 16 2011

Keywords

Comments

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a reflection or glide reflection in a line in one fixed direction perpendicular to the sides of the hexagons.

References

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

Crossrefs

A197554 Number of free poly-IH14-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 7, 24, 98, 415, 1846, 8378, 38764, 181482, 858387, 4091763, 19635210, 94749398
Offset: 1

Views

Author

Joseph Myers, Oct 16 2011

Keywords

Comments

Equivalently, polyhexes where two polyhexes are considered equivalent if and only if they are related by a translation or a reflection or glide reflection in a line in one fixed direction parallel to the sides of the hexagons.

References

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

Crossrefs

A056783 Number of diamond polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 62, 204, 709, 2526, 9212, 33989, 126838, 476597, 1802618, 6850969, 26153537, 100207548, 385225375, 1485216987, 5741272625, 22246000726, 86383442996, 336093551268, 1309998354125, 5114452295933, 19998173607505, 78306014924606, 307022185565345
Offset: 1

Views

Author

James Sellers, Aug 28 2000

Keywords

Comments

Also the number of polybricks of size n made of Lego.

Crossrefs

Formula

a(n) = 2*A006749(n) + A006746(n) + 2*A006748(n) + 2*A006747(n) + A056877(n) + 2*A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Extensions

More terms from Don Reble, Nov 01 2001
a(15)-a(18) from Joseph Myers, Nov 15 2010
Offset corrected and a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 24, 28, 33, 38, 43, 49, 55, 61, 68, 75, 82, 90, 97, 106, 114, 123, 133, 142, 152, 162, 173, 184, 195, 207, 219, 231, 244, 257, 270, 284, 297, 312, 326, 341, 357, 372, 388, 404, 421, 438, 455, 473, 491, 509, 528, 547, 566
Offset: 0

Views

Author

Jonathan Vos Post, Feb 27 2007

Keywords

Comments

Kurz proved the polyomino equivalent of this conjecture as A122133 and abstracts: "In this article we prove a conjecture of Bezdek, Brass and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each n."

Examples

			a(10) = 24 because floor((10^2 + 14*10/3 + 1)/6) = floor(24.6111111) = 24.
		

Crossrefs

Programs

  • Mathematica
    Table[Floor[(n^2+14n/3+1)/6],{n,0,80}] (* Harvey P. Dale, Apr 11 2012 *)
  • PARI
    concat(0, Vec(x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)) + O(x^50))) \\ Colin Barker, Oct 13 2016
    
  • PARI
    a(n) = (n^2 + 14*n/3 + 1)\6 \\ Charles R Greathouse IV, Oct 13 2016

Formula

a(n) = floor((n^2 + 14*n/3 + 1)/6).
G.f.: x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)). - Colin Barker, Oct 13 2016

Extensions

More terms from Harvey P. Dale, Apr 11 2012
Offset changed to 0 by Colin Barker, Oct 13 2016
Showing 1-9 of 9 results.