A258018 -id:A258018 - cp's OEIS frontend

A258015 Capped binary boundary codes for those fusenes that stay same when flipped over, only the maximal representative from each equivalence class up to rotation.

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, 8238832

Offset: 0

Antti Karttunen, Jun 01 2015

These are binary boundary codes for fusenes with bilateral symmetry, i.e., those terms k in A258013 for which A256999(A059893(k)) = k. A258018(n) gives the count of terms with binary width 2n + 1.

Differs from its subsequence A258005 for the first time at n=113, as a(113) = 131821024 is the first term not present in A258005.

Antti Karttunen, Table of n, a(n) for n = 0..113

Subsequence of A258013.
Subsequence: A258005.
Cf. A258018.

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

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 313

Offset: 1

Antti Karttunen, Jun 02 2015

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.

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).

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. A108070, A258017, A258018, A258206.

(define (A258019 n) (* (/ 1 2) (+ (A258017 n) (A258018 n))))

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)].
Other observations:
For all n, a(n) >= A258206(n).

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).

cp's OEIS Frontend

A258015 Capped binary boundary codes for those fusenes that stay same when flipped over, only the maximal representative from each equivalence class up to rotation.

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

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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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Programs

Scheme

Formula

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

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