A258003 -id:A258003 - cp's OEIS frontend

A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 26, 28, 30, 31, 32, 48, 52, 56, 58, 60, 62, 63, 64, 96, 100, 104, 106, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 192, 200, 208, 212, 224, 226, 228, 232, 234, 236, 240, 242, 244, 246, 248, 250, 252, 254, 255, 256, 384, 392, 400, 416, 420, 424, 426, 448, 450

Offset: 0

Antti Karttunen, May 16 2015

These correspond to the maximal (lexicographically largest) representatives selected from each equivalence class of binary necklaces. See the last example.
Indexing starts from zero, because a(0) = 0 is a special case.
If k is a member then so also is 2*k, i.e., k with 0 appended to the end of its binary representation.
If k is a member then so also is A004755(k), i.e., k with 1 prepended to the front of its binary representation.
One obtains A065609 if one erases the most significant bit of each term [as A053645(a(n))] and then discards any zero-terms produced from the terms that originally were powers of two (A000079).
First differs from A328607 in lacking 108, with binary expansion 1101100. If we define a dual-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is a dual-necklace. - Gus Wiseman, Nov 04 2019

			For n = 5, with binary representation "101", if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction, we get "110" = 6 > 5, so 5 can be made larger by such rotations, and thus is NOT included in this sequence.
For n = 6, with binary representation "110", no such rotation will yield a larger number, and thus 6 is included in this sequence.
For n = 28, with binary representation "11100", if we rotate non-msb bits towards right, we get additional numbers 22, 19 and 25 (with binary representations "10110", "10011", "11001") before coming to 28 again, and 28 is the largest of these numbers, thus 28 is included in this sequence.
  Also, if we discard the most significant bit of each and consider them just as binary strings, then A053645(28) = 12 is the lexicographically largest representative of {"1100", "0110", "0011", "1001"}, which is the complete set of representatives for a particular equivalence class of binary necklaces, obtained by rotating all bits of binary string "1100" successively towards right or left.

Antti Karttunen, Table of n, a(n) for n = 0..16637
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Complement: A257739.
Odd terms: A000225.
Subsequence of A065609.
Cf. A004755, A053645, A080541, A080542, A256999.
Subsequence: A258003.
The non-dual version is A328668.
The version involving all digits is A065609.
The non-dual reversed version is A328607.
Numbers whose reversed binary expansion is a necklace are A328595.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A069010, A275692, A328594, A328596.

reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[0,110],#<=1||reckQ[Rest[IntegerDigits[#,2]]]&] (* Gus Wiseman, Nov 04 2019 *)

A258013 Capped binary boundary codes for fusenes, only the maximal representatives of each equivalence class obtained by rotating.

Original entry on oeis.org

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 31 2015

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.

Antti Karttunen, Table of n, a(n) for n = 0..875
Eric Weisstein's World of Mathematics, Fusene
Wikipedia, Helicene

Subsequences: A258003, A258015.
Intersection of A257250 and A258012.
Cf. A258014 (same codes without the most significant bit).
Cf. also A258017.

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).

A258002 Capped binary boundary codes for holeless strictly non-overlapping polyhexes (all orientations and rotations included).

Original entry on oeis.org

1, 127, 1519, 1783, 1915, 1981, 2014, 6007, 7099, 7645, 7918, 20335, 22447, 23479, 23503, 23995, 24187, 24253, 24286, 26551, 27607, 28123, 28135, 28381, 28477, 28510, 29659, 30187, 30445, 30451, 30574, 30622, 31213, 31477, 31606, 31609, 31990, 32122, 32188, 80815, 81271, 89527, 89551, 89719, 93655, 93883, 95191, 95707, 95719, 95965, 96061

Offset: 0

Antti Karttunen, May 16 2015

The sequence consists of those terms of A255571 whose every A080541/A080542-rotation is also a term of A255571 and in their binary representation the number of 1's is larger than the number of 0's. More precisely, after the initial term a(0)=1 (which stands for an empty path) each term has seven more 1's than 0's in their binary representation, i.e., A037861(a(n)) = -7 for all n >= 1.

			8167737748888 is included in the sequence, as it encodes a 42-edge polyhex pattern which is composed of two seven-hex "crowns" connected by a snake-like "S-piece".

Antti Karttunen, Table of n, a(n) for n = 0..20622; all terms up to the binary width 26+1, solutions up to 26 edges

Intersection of A072600 and A258001.
Intersection of A255571 and A258012.
Subsequence: A258003 (lexicographically largest representatives).
Cf. A037861.
Differs from A258012 for the first time at n=6622.

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)).

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.

cp's OEIS Frontend

A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

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A258013 Capped binary boundary codes for fusenes, only the maximal representatives of each equivalence class obtained by rotating.

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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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A258002 Capped binary boundary codes for holeless strictly non-overlapping polyhexes (all orientations and rotations included).

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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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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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