cp's OEIS Frontend

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

%I A258019 #25 Feb 16 2025 08:33:25
%S A258019 0,0,1,0,1,1,3,2,12,14,50,97,313
%N A258019 Number of fusenes (not necessarily planar) of perimeter 2n, counted up to rotations and turning over.
%C A258019 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.
%C A258019 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).
%H A258019 Guo, Hansen, Zheng, <a href="http://dx.doi.org/10.1016/S0166-218X(01)00180-9">Boundary uniqueness of fusenes</a>, Discrete Applied Mathematics 118 (2002), pp. 209-222.
%H A258019 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fusene.html">Fusene</a>
%H A258019 Wikipedia, <a href="http://en.wikipedia.org/wiki/Helicene">Helicene</a>
%F A258019 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)].
%F A258019 Other observations:
%F A258019 For all n, a(n) >= A258206(n).
%o A258019 (Scheme) (define (A258019 n) (* (/ 1 2) (+ (A258017 n) (A258018 n))))
%Y A258019 Cf. A108070, A258017, A258018, A258206.
%K A258019 nonn,more
%O A258019 1,7
%A A258019 _Antti Karttunen_, Jun 02 2015