A258014 -id:A258014

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A258013

Capped binary boundary codes for fusenes, only the maximal representatives of each equivalence class obtained by rotating.

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

A258017(n) gives the count of terms with binary width 2n + 1.

Differs from A258003 for the first time at n=875, which here contains a(875) = 131821024 the smallest polyhex (26 edges, six hexes) where two hexes (at the opposite ends of a coiled pattern) meet to touch each other.

This pattern is isomorphic to benzenoid [6]Helicene (up to chirality, see the illustrations at Wikipedia-page).

Note that here, in contrast to "Boundary Edges Code for Benzenoid Systems" (see links at A258012), if a fusene has no bilateral symmetry then both variants of the corresponding one-sided fusene (their codes) are included in this sequence, the other obtained from the other by turning it over.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..875

Eric Weisstein's World of Mathematics, Fusene

Wikipedia, Helicene

CROSSREFS

Subsequences: A258003, A258015.

Intersection of A257250 and A258012.

Cf. A258014 (same codes without the most significant bit).

Cf. also A258017.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, May 31 2015

STATUS

approved

A258004

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

+10
4

0, 63, 990, 3822, 15222, 15738, 15804, 60858, 62394, 62940, 224694, 241110, 241338, 243162, 243420, 244188, 249306, 249318, 249564, 249660, 251370, 251628, 251634, 252396, 252660, 252792, 964314, 964326, 965340, 972522, 972636, 973548, 976620, 990678, 995034, 995046, 997098, 997212, 998124, 998130, 1003242, 1005420

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

FORMULA

a(n) = A053645(A258003(n)).

EXAMPLE

63 ("111111" in binary) is present as it encodes a single hex. This is because when we walk in honeycomb-lattice from vertex to vertex, at each vertex turning to the same direction, we will return to the starting vertex after enclosing a hex with six such steps.

PROG

(Scheme)

(define (A258004 n) (A053645 (A258003 n)))

CROSSREFS

Cf. A053645, A258003.

Subsequence of A255561 and A258014.