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)

Differs from A057779 for the first time at n=12 as here a(12) = 97, one less than A057779(12) because this sequence excludes polyhexes with holes, the smallest which contains six hexagons in a ring, enclosing a hole of one hex, having thus perimeter of 18+6 = 24 (= 2*12) edges.

Differs from A258019 for the first time at n=13 as here a(13) = 312, one less than A258019(13) because this sequence counts only strictly non-overlapping and non-touching polyhex-patterns, while A258019(13) already includes one specimen of helicene-like self-reaching structures.

If one counts these structures by the number of hexagons (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 81, ... (A018190).

a(n) is also the number of 2n-step 2-dimensional closed self-avoiding paths on honeycomb lattice, reduced for symmetry. - Luca Petrone, Jan 08 2016

This sequence counts fusenes up to rotations, but with no turning over allowed. Fusenes are like polyhexes with additional criteria that no holes are allowed, while on the other hand, helicene-like self-touching or self-overlapping configurations are included in the count here. Cf. the links and further comments at A258019.

For n >= 1, a(n) gives the total number of terms k in A258013 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

This sequence counts fusenes which stay the same when flipped over. Fusenes are like polyhexes with additional criteria that no holes are allowed, but on the other hand, helicene-like self-touching or self-overlapping configurations are included in the count here. Cf. the links and further comments at A258019.

For n >= 1, a(n) gives the total number of terms k in A258015 with binary width = 2n + 1, or equally, with A000523(k) = 2n.