cp's OEIS Frontend

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

%I A258206 #26 May 12 2025 03:09:46
%S A258206 0,0,1,0,1,1,3,2,12,14,50,97,312,744,2291,6186,18714,53793,162565,
%T A258206 482416,1467094,4436536,13594266,41640513,128564463,397590126,
%U A258206 1236177615,3852339237,12053032356,37802482958,118936687722,375079338476
%N A258206 Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.
%C A258206 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.
%C A258206 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.
%C A258206 If one counts these structures by the number of hexagons (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 81, ... (A018190).
%C A258206 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
%D A258206 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.
%H A258206 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a258206.htm">Illustration of polygons of perimeter <= 20</a>.
%F A258206 a(n) = (1/2) * (A258204(n) + A258205(n)).
%F A258206 Other observations. For all n >= 1:
%F A258206 a(n) <= A057779(n).
%F A258206 a(n) <= A258019(n).
%o A258206 (Scheme) (define (A258206 n) (* (/ 1 2) (+ (A258204 n) (A258205 n))))
%Y A258206 Cf. A000228, A057779, A018190, A258019, A258204, A258205, A316193.
%K A258206 nonn,walk,more
%O A258206 1,7
%A A258206 _Antti Karttunen_, May 31 2015
%E A258206 a(14)-a(15) from _Luca Petrone_, Jan 08 2016
%E A258206 a(16)-a(23) from Cyvin, Brunvoll & Cyvin added by _Andrey Zabolotskiy_, Mar 01 2023
%E A258206 a(24)-a(32) from _Bert Dobbelaere_, May 12 2025