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A258019 Number of fusenes (not necessarily planar) of perimeter 2n, counted up to rotations and turning over. 5
0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 313 (list; graph; refs; listen; history; text; internal format)



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).


Table of n, a(n) for n=1..13.

Guo, Hansen, Zheng, Boundary uniqueness of fusenes, Discrete Applied Mathematics 118 (2002), pp. 209-222.

Eric Weisstein's World of Mathematics, Fusene

Wikipedia, Helicene


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).


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


Cf. A108070, A258017, A258018, A258206.

Sequence in context: A129925 A258206 A267011 * A057779 A005220 A243660

Adjacent sequences:  A258016 A258017 A258018 * A258020 A258021 A258022




Antti Karttunen, Jun 02 2015



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Last modified July 20 14:23 EDT 2019. Contains 325185 sequences. (Running on oeis4.)