A258205 -id:A258205 - 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

A258204 Number of one-sided strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotation.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 16, 23, 80, 183, 563

Offset: 1

Antti Karttunen, May 31 2015

For n >= 1, a(n) gives the total number of terms k in A258003 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

Cf. A057779, A258003, A258017, A258205, A258206.

(define (A258204 n) (let loop ((k (+ 1 (expt 2 (+ n n)))) (c 0)) (cond ((pow2? k) c) (else (loop (+ 1 k) (+ c (if (isA258003? 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 isA258003? given in A258003.

Other identities and observations. For all n >= 1:
a(n) = 2*A258206(n) - A258205(n).
a(n) <= A258017(n).

A258005 Capped binary boundary codes for holeless strictly non-overlapping polyhexes with bilateral symmetry, only the maximal representative from each equivalence class obtained by rotating.

Original entry on oeis.org

1, 127, 2014, 7918, 31606, 32122, 32188, 126394, 486838, 503482, 505564, 506332, 511708, 511804, 513514, 514936, 2012890, 2021098, 2025196, 2054044, 2055544, 7788250, 8050522, 8051434, 8051548, 8054620, 8075098, 8075110, 8084380, 8104888, 8182636, 8183020, 8185756, 8207218, 8207602, 8214442, 8219596, 8219602, 8231884, 8236516

Offset: 0

Antti Karttunen, May 31 2015

Indexing starts from zero, because a(0) = 1 is a special case, indicating an empty path in the honeycomb lattice.
These are capped binary boundary codes for those holeless polyhexes that stay same when they are flipped over and rotated appropriately.
A258205(n) gives the count of terms with binary width 2n + 1.

Antti Karttunen, Table of n, a(n) for n = 0..112

Cf. A255571, A258205.
Intersection of A258003 and A258209. Differs from A258003 for the first time at n=8, where a(8) = 486838 while A258003(8) = 127930.
Subsequence of A258015 from which this differs for the first time at n=113.

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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A258204 Number of one-sided strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotation.

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A258005 Capped binary boundary codes for holeless strictly non-overlapping polyhexes with bilateral symmetry, only the maximal representative from each equivalence class obtained by rotating.

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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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