A258204 -id:A258204 - cp's OEIS frontend

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

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291, 6186, 18714, 53793, 162565, 482416, 1467094, 4436536, 13594266, 41640513, 128564463, 397590126, 1236177615, 3852339237, 12053032356, 37802482958, 118936687722, 375079338476

Offset: 1

Antti Karttunen, May 31 2015

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.
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.
If one counts these structures by the number of hexagons (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 81, ... (A018190).
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

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.

Hugo Pfoertner, Illustration of polygons of perimeter <= 20.

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

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

a(n) = (1/2) * (A258204(n) + A258205(n)).
Other observations. For all n >= 1:
a(n) <= A057779(n).
a(n) <= A258019(n).

a(14)-a(15) from Luca Petrone, Jan 08 2016
a(16)-a(23) from Cyvin, Brunvoll & Cyvin added by Andrey Zabolotskiy, Mar 01 2023
a(24)-a(32) from Bert Dobbelaere, May 12 2025

A258003 Capped binary boundary codes for holeless strictly non-overlapping polyhexes, only the maximal representative from each equivalence class obtained by rotating.

Original entry on oeis.org

1, 127, 2014, 7918, 31606, 32122, 32188, 126394, 127930, 128476, 486838, 503254, 503482, 505306, 505564, 506332, 511450, 511462, 511708, 511804, 513514, 513772, 513778, 514540, 514804, 514936, 2012890, 2012902, 2013916, 2021098, 2021212, 2022124, 2025196, 2039254, 2043610, 2043622, 2045674, 2045788, 2046700

Offset: 0

Antti Karttunen, May 16 2015

Indexing starts from zero, because a(0) = 1 is a special case, indicating an empty path, which thus ends at the same vertex as where it started from.

A258204(n) gives the count of terms with binary width 2n + 1.

Antti Karttunen, Table of n, a(n) for n = 0..874

Intersection of A257250 and A258002.
Subsequence of A258013.
Subsequence: A258005.
Cf. A037861, A080541, A080542, A255571, A256999, A258204.
Cf. also A258004 (the same terms without the most significant bit, slightly more compact representation).

A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 1, 8, 5, 20, 11, 61

Offset: 1

Antti Karttunen, May 31 2015

This sequence counts by perimeter length those holeless polyhexes that stay same when they are flipped over and rotated appropriately.

For n >= 1, a(n) gives the total number of terms k in A258005 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

Cf. A057779, A258005, A258018, A258204, A258206.

(define (A258205 n) (let loop ((k (+ 1 (expt 2 (+ n n)))) (c 0)) (cond ((pow2? k) c) (else (loop (+ 1 k) (+ c (if (isA258005? k) 1 0)))))))
(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i)))))) ;; Gives non-false only when n is a power of two.
;; Code for isA258005? given in A258005.

Other identities and observations. For all n >= 1:
a(n) = 2*A258206(n) - A258204(n).
a(n) <= A258018(n).

A258017 Number of one-sided fusenes (not necessarily planar) of perimeter 2n, counted up to rotations.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 16, 23, 80, 183, 564

Offset: 1

Antti Karttunen, Jun 02 2015

This sequence counts fusenes up to rotations, but with no turning over allowed. Fusenes are like polyhexes with additional criteria that no holes are allowed, while on the other hand, helicene-like self-touching or self-overlapping configurations are included in the count here. Cf. the links and further comments at A258019.

For n >= 1, a(n) gives the total number of terms k in A258013 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

Xiaofeng Guo, Pierre Hansen, and Maolin Zheng, Boundary uniqueness of fusenes, Discrete Applied Mathematics 118 (2002), pp. 209-222.
Pornrat Ruengrot and Duangkamon Baowan, Classification of k-defect holes on a graphene sheet, Comp. Materials Sci. (2023) Vol. 225, 112181.
Eric Weisstein's World of Mathematics, Fusene
Wikipedia, Helicene

Cf. A258018, A258019, A258204.

Cf. A258013.

Other identities and observations. For all n >= 1:
a(n) = 2*A258019(n) - A258018(n).
a(n) >= A258204(n).

A258018 Number of fusenes of perimeter 2n (not necessarily planar) with bilateral symmetry, counted up to rotations.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 1, 8, 5, 20, 11, 62

Offset: 1

Antti Karttunen, Jun 02 2015

This sequence counts fusenes which stay the same when flipped over. Fusenes are like polyhexes with additional criteria that no holes are allowed, but on the other hand, helicene-like self-touching or self-overlapping configurations are included in the count here. Cf. the links and further comments at A258019.

For n >= 1, a(n) gives the total number of terms k in A258015 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

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

Cf. A258017, A258019, A258204.

Cf. A258015.

Other identities and observations. For all n >= 1:
a(n) = 2*A258019(n) - A258017(n).
a(n) >= A258205(n).

cp's OEIS Frontend

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

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A258003 Capped binary boundary codes for holeless strictly non-overlapping polyhexes, only the maximal representative from each equivalence class obtained by rotating.

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A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

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A258017 Number of one-sided fusenes (not necessarily planar) of perimeter 2n, counted up to rotations.

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A258018 Number of fusenes of perimeter 2n (not necessarily planar) with bilateral symmetry, counted up to rotations.

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