A258206 - OEIS

%I #22 Mar 01 2023 11:05:11

%S 0,0,1,0,1,1,3,2,12,14,50,97,312,744,2291,6186,18714,53793,162565,

%T 482416,1467094,4436536,13594266

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

%C 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 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 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 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 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 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a258206.htm">Illustration of polygons of perimeter <= 20</a>.

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

%F Other observations. For all n >= 1:

%F a(n) <= A057779(n).

%F a(n) <= A258019(n).

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

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

%K nonn,walk,more

%O 1,7

%A _Antti Karttunen_, May 31 2015

%E a(14)-a(15) from _Luca Petrone_, Jan 08 2016

%E a(16)-a(23) from Cyvin, Brunvoll & Cyvin added by _Andrey Zabolotskiy_, Mar 01 2023