A255561 -id:A255561 - cp's OEIS frontend

A255571 Numbers whose binary representation traces a nonselfcrossing circuit in honeycomb lattice when its bits (from the least to the second most significant bit) are interpreted as directions to proceed at each vertex. (The most significant 1-bit is ignored).

1, 64, 65, 126, 127, 1056, 1057, 1090, 1091, 1156, 1157, 1288, 1289, 1518, 1519, 1552, 1553, 1782, 1783, 1914, 1915, 1980, 1981, 2014, 2015, 4368, 4369, 4642, 4643, 5188, 5189, 6006, 6007, 6280, 6281, 7098, 7099, 7644, 7645, 7918, 7919, 16962, 16963, 17028, 17029, 17160, 17161, 17542, 17543, 17544, 17545, 17674, 17675, 17938, 17939

Offset: 0

Antti Karttunen, Apr 13 2015

Numbers n such that when we start scanning bits in the binary expansion of n, from the least to the most 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 except the most significant 1-bit (which is ignored), 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) = 1 is a special case, indicating an empty path, which thus ends at the same vertex as where it started from.
If n is a member, then A054429(n) is also a member.

			The examples given in A255570 occur also in this sequence, except that 380 ("101111100" in binary) is excluded from this sequence, because it visits twice the first vertex after the starting vertex.

Antti Karttunen, Table of n, a(n) for n = 0..9300
Wikipedia, Hexagonal lattice

Cf. A054429, A255561.

Subsequence of A255570.

A254109 If n <= 63, a(n) = n; for n > 63: a(32n + 14) = 8n + 5, a(64n + 30) = 4n + 3, and for other cases with n > 63: a(2n) = 2a(n), a(2n+1) = 2a(n) + 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 21, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 7

Offset: 0

Antti Karttunen, Mar 11 2015

This sequence is a rewriting-recurrence which attempts to contract the perimeter of binary boundary coded holeless polyhexes and other fusenes by 2 or 4 edges, where first possible (from the least significant end of n), and if no such contraction is possible, then it fixes n. Together with recurrence A258009 can be used to obtain the terms of A258012, please see comments there.

			The first term where a(n) is different from n occurs at n=78, as 78 = "1001110" in binary, where the clause a(32n + 14) = 8*n + 5 will rewrite the trailing "01110" part as "101", resulting binary string "10101" = 21 in decimal.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A255561, A255568, A258009, A258012.

If n <= 63, a(n) = n; for n > 63: a(32n + 14) = 8*n + 5, a(64n + 30) = 4*n + 3, and for other cases with n > 63: a(2n) = 2*a(n), a(2n+1) = 2*a(n) + 1.

Recurrence corrected to match the intended usage by Antti Karttunen, Jun 05 2015

A258004 Capless 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

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, May 16 2015

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.

			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.

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

Cf. A053645, A258003.

Subsequence of A255561 and A258014.

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

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

A344022 Numbers with binary expansion (b_1, ..., b_m) such that bending a strip of paper of length k+1 with an angle of +90 degrees (resp. -90 degrees) at position X=k when b_k = 1 (resp. b_k = 0) for k = 1..m yields a configuration where all edges are distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85

Offset: 1

Rémy Sigrist, May 07 2021

All positive terms belong to A166535, but the reverse is not true (for example, A166535(96) = 136 does not belong to this sequence).
This sequence is infinite as it contains A000975 and A343183.
If m belongs to the sequence, then floor(m/2) also belongs to the sequence.
For any k > 0, the sequence contains A006744(k) positive terms with k binary digits.
This sequence has connections with A258002, A255561 and A255571: these sequences encode in binary nonoverlapping or noncrossing paths in the honeycomb lattice.

			See illustration in Links section.

Rémy Sigrist, Illustration of initial terms

Cf. A000975, A006744, A166535, A258002, A255561, A255571, A343183, A344145.

is(n) = { my (b=binary(n), d=1, s=[d], z=2*d); for (k=1, #b, if (b[k], d*=I, d/=I); if (setsearch(s, z+=d), return (0), s=setunion(s, [z]); z+=d)); return (1) }

cp's OEIS Frontend

A255571 Numbers whose binary representation traces a nonselfcrossing circuit in honeycomb lattice when its bits (from the least to the second most significant bit) are interpreted as directions to proceed at each vertex. (The most significant 1-bit is ignored).

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A254109 If n <= 63, a(n) = n; for n > 63: a(32n + 14) = 8n + 5, a(64n + 30) = 4n + 3, and for other cases with n > 63: a(2n) = 2a(n), a(2n+1) = 2a(n) + 1.

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A258004 Capless binary boundary codes for holeless strictly non-overlapping polyhexes, only the maximal representative from each equivalence class obtained by rotating.

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A344022 Numbers with binary expansion (b_1, ..., b_m) such that bending a strip of paper of length k+1 with an angle of +90 degrees (resp. -90 degrees) at position X=k when b_k = 1 (resp. b_k = 0) for k = 1..m yields a configuration where all edges are distinct.

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PARI

cp's OEIS Frontend

A255571 Numbers whose binary representation traces a nonselfcrossing circuit in honeycomb lattice when its bits (from the least to the second most significant bit) are interpreted as directions to proceed at each vertex. (The most significant 1-bit is ignored).

Views

Author

Keywords

Comments

Examples

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Crossrefs

A254109 If n <= 63, a(n) = n; for n > 63: a(32n + 14) = 8*n + 5, a(64n + 30) = 4*n + 3, and for other cases with n > 63: a(2n) = 2*a(n), a(2n+1) = 2*a(n) + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A258004 Capless binary boundary codes for holeless strictly non-overlapping polyhexes, only the maximal representative from each equivalence class obtained by rotating.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A344022 Numbers with binary expansion (b_1, ..., b_m) such that bending a strip of paper of length k+1 with an angle of +90 degrees (resp. -90 degrees) at position X=k when b_k = 1 (resp. b_k = 0) for k = 1..m yields a configuration where all edges are distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A254109 If n <= 63, a(n) = n; for n > 63: a(32n + 14) = 8n + 5, a(64n + 30) = 4n + 3, and for other cases with n > 63: a(2n) = 2a(n), a(2n+1) = 2a(n) + 1.