cp's OEIS Frontend

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)

These are polyominoes of the Euclidean regular tiling of hexagons with Schläfli symbol {6,3}. This sequence can most readily be calculated by enumerating fixed polyominoes for three situations: 1) fixed polyominoes with a horizontal axis of symmetry along an edge of a cell with no cell centered on that axis, A001207(n/2), 2) fixed polyominoes with a horizontal axis of symmetry that is a diagonal of at least one cell, A347258, and 3) fixed polyominoes with a horizontal axis of symmetry that joins the midpoints of opposite edges of at least one cell, A347257. These three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Aug 24 2021

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

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.

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.

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.

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.

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.

These are polyominoes of the Euclidean hexagonal regular tiling with Schläfli symbol {6,3}. This is one of three sequences needed to calculate the number of achiral polyominoes, A030225. The three sequences together contain exactly two copies of each achiral polyomino. This sequence can be calculated using a modification of Redelmeier's method; one chooses an original cell that is leftmost on and bisected by the axis of symmetry along a horizontal line connecting midpoints of opposite edges of one cell. Neighbors are added only if their centers are above the axis of symmetry or on the axis of symmetry to the right of the original cell. Cells not centered on the axis of symmetry are counted twice to include their reflections.