A258004 -id:A258004 - cp's OEIS frontend

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

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).

A255561 Numbers whose binary representation traces a non-selfcrossing circuit in the honeycomb lattice when each one of its bits, from the most significant to the least significant, is interpreted as a direction to proceed at each vertex.

Original entry on oeis.org

0, 32, 63, 528, 545, 578, 644, 759, 776, 891, 957, 990, 1007, 2184, 2321, 2594, 3003, 3140, 3549, 3822, 3959, 8481, 8514, 8580, 8771, 8772, 8837, 8969, 9264, 9288, 9350, 9353, 9482, 9746, 10167, 10320, 10337, 10385, 10508, 10514, 10772, 11223, 11300, 11739, 11751, 11997, 12093, 12126, 12143, 12432, 12449, 12482, 12578, 12824, 12836, 13275

Offset: 0

Antti Karttunen, Apr 13 2015

Numbers n such that when we start scanning bits in the binary expansion of n, from the most to the least significant end, and when we interpret each bit as to a direction which to turn at each vertex (e.g., 0 = left, 1 = right) when traversing the edges of honeycomb lattice, then, when we have consumed all the bits, we have eventually returned to the same vertex where we started from and none of the other vertices have been visited twice.

Indexing starts from zero, because a(0) = 0 is a special case, indicating an empty path, which certainly ends at the same vertex as where it starts from. For other cases, we always take first a right turn, because the most significant bit is always 1.

			32 ("100000" in binary) is included, because if we take first turn to the right at some vertex, and then five turns to the left in succession, we will reach the same vertex where we started from.
63 ("111111" in binary) is included, because when we take six turns to the right in the hexagonal lattice, we will reach the same vertex where we started from.
528 ("1000010000" in binary) is included, because it traces the edges of two adjacent hexagons, returning to the same vertex where the path started from, which is the other of the two vertices shared by those hexagons.

Antti Karttunen, Table of n, a(n) for n = 0..13242
Wikipedia, Hexagonal lattice

Cf. A014486, A255571.

Subsequence: A258004.

A258014 Capless binary boundary codes for fusenes, maximal representative from each equivalence class up to rotation.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 01 2015

Differs from A258004 for the first time at n=875, which here contains a(875) = 64712160, encoding the smallest polyhex (26 edges, six hexes) where two hexes (at the different ends of the pattern) meet to touch each other.

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

Cf. A053645, A258013.

Cf. A258004 (a subsequence).

(define (A258014 n) (A053645 (A258013 n)))

a(n) = A053645(A258013(n)).

cp's OEIS Frontend

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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A255561 Numbers whose binary representation traces a non-selfcrossing circuit in the honeycomb lattice when each one of its bits, from the most significant to the least significant, is interpreted as a direction to proceed at each vertex.

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A258014 Capless binary boundary codes for fusenes, maximal representative from each equivalence class up to rotation.

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