A258019 Number of fusenes (not necessarily planar) of perimeter 2n, counted up to rotations and turning over.

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 313

Offset: 1

Author

Antti Karttunen, Jun 02 2015

Keywords

• nonn
• more

Comments

A fusene is a benzenoid (a polyhex) which has a single component of boundary edges (that is, no holes). Including also geometrically nonplanar configurations allows helicene-like self-touching or self-overlapping structures. Thus this sequence differs from A258206 for the first time at n=13 as here a(13) = 313 [while A258206(13) = 312] because the smallest such nonplanar structure is 26-edge [6]Helicene, which is encoded by one-capped binary code 131821024 (= A258013(875) = A258015(113)). Please see the illustrations at the Wikipedia page. Note that although in their three-dimensional conformation molecules like [6]Helicene and other [n]Helicenes with n >= 6 have two different chiralities (resulting from the handedness of the helicity itself), in this count of abstract combinatorial objects they are considered achiral because of their bilateral symmetry.

If one counts these structures by the number of hexes (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 82, ... (probably A108070).

Links

• Guo, Hansen, Zheng, Boundary uniqueness of fusenes, Discrete Applied Mathematics 118 (2002), pp. 209-222.
• Eric Weisstein's World of Mathematics, Fusene
• Wikipedia, Helicene

Crossrefs

Cf. A108070, A258017, A258018, A258206.

Programs

Scheme

(define (A258019 n) (* (/ 1 2) (+ (A258017 n) (A258018 n))))

Formula

a(n) = (1/2) * (A258017(n) + A258018(n)). [1/2 times the count of one-sided fusenes + the count of fusenes with bilateral symmetry (subset of the former)].

Other observations:

For all n, a(n) >= A258206(n).