A258206 Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291, 6186, 18714, 53793, 162565, 482416, 1467094, 4436536, 13594266, 41640513, 128564463, 397590126, 1236177615, 3852339237, 12053032356, 37802482958, 118936687722, 375079338476

Offset: 1

Author

Antti Karttunen, May 31 2015

Keywords

• nonn
• walk
• more

Comments

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

References

• S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Theory of Coronoid Hydrocarbons, Springer-Verlag, 1991. See sections 4.7 Annulene and 6.5 Annulenes.

Links

• Hugo Pfoertner, Illustration of polygons of perimeter <= 20.

Crossrefs

Cf. A000228, A057779, A018190, A258019, A258204, A258205, A316193.

Programs

Scheme

(define (A258206 n) (* (/ 1 2) (+ (A258204 n) (A258205 n))))

Formula

a(n) = (1/2) * (A258204(n) + A258205(n)).

Other observations. For all n >= 1:

a(n) <= A057779(n).

a(n) <= A258019(n).

Extensions

a(14)-a(15) from Luca Petrone, Jan 08 2016

a(16)-a(23) from Cyvin, Brunvoll & Cyvin added by Andrey Zabolotskiy, Mar 01 2023

a(24)-a(32) from Bert Dobbelaere, May 12 2025