A211100 -id:A211100 - cp's OEIS frontend

A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 2, 4, 6, 4, 0, 0, 4, 6, 9, 8, 5, 0, 0, 2, 6, 15, 20, 15, 6, 0, 0, 4, 8, 24, 32, 35, 18, 7, 0, 0, 3, 10, 27, 56, 70, 54, 28, 8, 0, 0, 4, 12, 42, 84, 125, 120, 84, 32, 9, 0, 0, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Gus Wiseman, Apr 16 2020

A composition of n is a finite sequence of positive integers summing to n.

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   3   2   3   0
   0   2   4   6   4   0
   0   4   6   9   8   5   0
   0   2   6  15  20  15   6   0
   0   4   8  24  32  35  18   7   0
   0   3  10  27  56  70  54  28   8   0
   0   4  12  42  84 125 120  84  32   9   0
   0   2  10  45 120 210 252 210 120  45  10   0
   0   6  18  66 168 335 450 462 320 162  50  11   0
Row n = 6 counts the following compositions (empty columns indicated by dots):
  .  (6)       (15)    (114)  (1113)  (11112)  .
     (33)      (24)    (123)  (1122)  (11121)
     (222)     (42)    (132)  (1131)  (11211)
     (111111)  (51)    (141)  (1221)  (12111)
               (1212)  (213)  (1311)  (21111)
               (2121)  (231)  (2112)
                       (312)  (2211)
                       (321)  (3111)
                       (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column k = 1 is A000005.
Row sums are A011782.
Diagonal T(2n,n) is A045630(n).
The strict version is A072574.
A version counting runs is A238279.
Column k = n - 1 is A254667.
Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Period of binary expansion is A302291.
Numbers whose prime signature is aperiodic are A329139.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Rotational symmetries are counted by A138904.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A291166, A328595, A328596, A329312, A329313, A329326.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[c,Length[Union[Array[RotateRight[c,#]&,Length[c]]]]==k]]],{n,0,10},{k,0,n}]

T(n,k)=if(n==0, k==0, sumdiv(n, m, sumdiv(gcd(k,m), d, moebius(d)*binomial(m/d-1, k/d-1)))) \\ Andrew Howroyd, Jan 19 2023

T(n,k) = Sum_{m|n} Sum_{d|gcd(k,m)} mu(d)*binomial(m/d-1, k/d-1) for n > 0. - Andrew Howroyd, Jan 19 2023

A211096 Smallest (i.e., rightmost) Lyndon word in the Lyndon factorization of the binary representation of n (written using 1's and 2's rather than 0's and 1's, since numbers > 0 in the OEIS cannot begin with 0).

Original entry on oeis.org

1, 2, 1, 2, 1, 12, 1, 2, 1, 112, 1, 122, 1, 12, 1, 2, 1, 1112, 1, 1122, 1, 12, 1, 1222, 1, 112, 1, 122, 1, 12, 1, 2, 1, 11112, 1, 11122, 1, 11212, 1, 11222, 1, 112, 1, 12122, 1, 12, 1, 12222, 1, 1112, 1, 1122, 1, 12, 1, 1222, 1, 112, 1, 122, 1, 12, 1, 2, 1, 111112, 1, 111122, 1, 111212, 1, 111222, 1, 112, 1, 112122, 1, 112212, 1, 112222, 1, 1112, 1, 1122, 1

Offset: 0

N. J. A. Sloane, Mar 31 2012

nonn

See A211095 and A211097 for further information, including Maple programs.

			n=25 has binary expansion 11001, which has Lyndon factorization (1)(1)(001) with three factors. The rightmost factor is 001, which we write as a(25) = 112.
The real sequence (written with 0's and 1's rather than 1's and 2's) is:
0, 1, 0, 1, 0, 01, 0, 1, 0, 001, 0, 011, 0, 01, 0, 1, 0, 0001, 0, 0011, 0, 01, 0, 0111, 0, 001, 0, 011, 0, 01, 0, 1, 0, 00001, 0, 00011, 0, 00101, 0, 00111, 0, 001, 0, 01011, 0, 01, 0, 01111, 0, 0001, 0, 0011, 0, 01, 0, 0111, 0, 001, 0, 011, ...

Cf. A211100, A211095, A211097, A211098, A211099.

a(2k) is always 1 (i.e., 0).

A334297 Length of the Lyndon factorization of the reversed n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2

Offset: 0

Gus Wiseman, Apr 25 2020

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 12345th composition is (1,7,1,1,3,1), with reverse (1,3,1,1,7,1), with Lyndon factorization ((1),(1,3),(1,1,7)), so a(12345) = 3.

The non-reversed version is A329312.
The version for binary indices is A329313 (also the "co-" version).
Positions of 1's are A334265 (reversed Lyndon words).
Binary Lyndon words are counted by A001037 and ranked by A102659.
Lyndon compositions are counted by A059966 and ranked by A275692.
Normal Lyndon sequences are counted by A060223 (row sums of A296372).
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Aperiodic compositions are A328594.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Length of co-Lyndon factorization is A334029.
Cf. A027375, A211097, A211100, A328595, A329314, A329317.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
Table[Length[lynfac[Reverse[stc[n]]]],{n,0,100}]

A329396 Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 38, 40, 42, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 136, 140, 142, 144, 160, 164, 168, 170, 192

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

A sequence of words is uniform if they all have the same length.

			The sequence of terms together with their co-Lyndon factorizations begins:
   1:      (1) = (1)
   2:     (10) = (10)
   3:     (11) = (1)(1)
   4:    (100) = (100)
   6:    (110) = (110)
   7:    (111) = (1)(1)(1)
   8:   (1000) = (1000)
  10:   (1010) = (10)(10)
  12:   (1100) = (1100)
  14:   (1110) = (1110)
  15:   (1111) = (1)(1)(1)(1)
  16:  (10000) = (10000)
  20:  (10100) = (10100)
  24:  (11000) = (11000)
  26:  (11010) = (11010)
  28:  (11100) = (11100)
  30:  (11110) = (11110)
  31:  (11111) = (1)(1)(1)(1)(1)
  32: (100000) = (100000)
  36: (100100) = (100)(100)
  38: (100110) = (100)(110)
  40: (101000) = (101000)
  42: (101010) = (10)(10)(10)

Numbers whose binary expansion has uniform Lyndon factorization are A023758.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.
Cf. A001037, A059966, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329398.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Select[Range[100],SameQ@@Length/@colynfac[IntegerDigits[#,2]]&]

A329399 Numbers whose reversed binary expansion has uniform Lyndon factorization.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 38, 40, 42, 44, 48, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 88, 92, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 136, 140, 142, 144, 152, 160, 164, 168, 170

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

A sequence of words is uniform if they all have the same length.

			The sequence of terms together with their reversed binary expansions and Lyndon factorizations begins:
   1:      (1) = (1)
   2:     (01) = (01)
   3:     (11) = (1)(1)
   4:    (001) = (001)
   6:    (011) = (011)
   7:    (111) = (1)(1)(1)
   8:   (0001) = (0001)
  10:   (0101) = (01)(01)
  12:   (0011) = (0011)
  14:   (0111) = (0111)
  15:   (1111) = (1)(1)(1)(1)
  16:  (00001) = (00001)
  20:  (00101) = (00101)
  24:  (00011) = (00011)
  26:  (01011) = (01011)
  28:  (00111) = (00111)
  30:  (01111) = (01111)
  31:  (11111) = (1)(1)(1)(1)(1)
  32: (000001) = (000001)
  36: (001001) = (001)(001)
  38: (011001) = (011)(001)
  40: (000101) = (000101)
  42: (010101) = (01)(01)(01)
  44: (001101) = (001101)
  48: (000011) = (000011)

Numbers whose binary expansion has uniform Lyndon factorization and uniform co-Lyndon factorization are A023758.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.
Cf. A001037, A059966, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329396.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
Select[Range[100],SameQ@@Length/@lynfac[Reverse[IntegerDigits[#,2]]]&]

A329400 Length of the co-Lyndon factorization of the binary expansion of n with the most significant (first) digit removed.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 3, 3, 2, 3, 1, 2, 1, 3, 4, 4, 3, 4, 2, 3, 2, 4, 1, 2, 2, 3, 1, 2, 1, 4, 5, 5, 4, 5, 3, 4, 3, 5, 2, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 6, 6, 5, 6, 4, 5, 4, 6, 3, 4, 4, 5, 3, 4, 3, 6, 2, 3, 3, 4, 2, 4, 3, 5

Offset: 1

Gus Wiseman, Nov 16 2019

nonn

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

			Decapitated binary expansions of 1..20 together with their co-Lyndon factorizations:
   1:     () =
   2:    (0) = (0)
   3:    (1) = (1)
   4:   (00) = (0)(0)
   5:   (01) = (0)(1)
   6:   (10) = (10)
   7:   (11) = (1)(1)
   8:  (000) = (0)(0)(0)
   9:  (001) = (0)(0)(1)
  10:  (010) = (0)(10)
  11:  (011) = (0)(1)(1)
  12:  (100) = (100)
  13:  (101) = (10)(1)
  14:  (110) = (110)
  15:  (111) = (1)(1)(1)
  16: (0000) = (0)(0)(0)(0)
  17: (0001) = (0)(0)(0)(1)
  18: (0010) = (0)(0)(10)
  19: (0011) = (0)(0)(1)(1)
  20: (0100) = (0)(100)

The non-"co" version is A211097.
The version involving all digits is A329312.
Lyndon and co-Lyndon compositions are (both) counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Numbers whose decapitated binary expansion is co-Lyndon are A329401.
Cf. A001037, A060223, A102659, A211100, A329313, A329318, A329326, A329395.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[If[n==0,0,Length[colynfac[Rest[IntegerDigits[n,2]]]]],{n,30}]

A329401 Numbers whose binary expansion without the most significant (first) digit is a co-Lyndon word.

Original entry on oeis.org

2, 3, 6, 12, 14, 24, 28, 30, 48, 52, 56, 58, 60, 62, 96, 104, 112, 114, 116, 120, 122, 124, 126, 192, 200, 208, 212, 224, 226, 228, 232, 234, 236, 240, 242, 244, 246, 248, 250, 252, 254, 384, 400, 416, 420, 424, 448, 450, 452, 456, 458, 464, 466, 468, 472, 474

Offset: 1

Gus Wiseman, Nov 16 2019

nonn

A co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations.

			The sequence of terms together with their binary expansions begins:
    2: (1,0)
    3: (1,1)
    6: (1,1,0)
   12: (1,1,0,0)
   14: (1,1,1,0)
   24: (1,1,0,0,0)
   28: (1,1,1,0,0)
   30: (1,1,1,1,0)
   48: (1,1,0,0,0,0)
   52: (1,1,0,1,0,0)
   56: (1,1,1,0,0,0)
   58: (1,1,1,0,1,0)
   60: (1,1,1,1,0,0)
   62: (1,1,1,1,1,0)
   96: (1,1,0,0,0,0,0)
  104: (1,1,0,1,0,0,0)
  112: (1,1,1,0,0,0,0)
  114: (1,1,1,0,0,1,0)
  116: (1,1,1,0,1,0,0)
  120: (1,1,1,1,0,0,0)

The version involving all digits is A275692.
Binary Lyndon/co-Lyndon words are A001037.
A ranking of binary co-Lyndon words is A329318
Cf. A059966, A060223, A102659, A211097, A211100, A328594, A328596, A329312, A329325, A329326, A329359, A329395, A329396, A329400.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[2,100],colynQ[Rest[IntegerDigits[#,2]]]&]

A211098 Length of largest (i.e., leftmost) Lyndon word in Lyndon factorization of binary vectors of lengths 1, 2, 3, ...

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 1, 4, 3, 4, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 5, 3, 5, 4, 5, 2, 2, 2, 5, 3, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 6, 4, 6, 5, 6, 3, 3, 5, 6, 4, 6, 5, 6, 2, 2, 2, 2, 2, 2, 5, 6, 3, 3, 3, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Apr 01 2012

nonn

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). Here we look at the Lyndon factorizations of the binary vectors 0,1, 00,01,10,11, 000,001,010,011,100,101,110,111, 0000,...
See A211097, A211099, A211100 for further information, including Maple code.
The smallest (or rightmost) factors are given by A211095 and A211096, offset by 2.

			Here are the Lyndon factorizations of the first few binary vectors:
.0.
.1.
.0.0.
.01.
.1.0.
.1.1.
.0.0.0.
.001.
.01.0.
.011.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.0.0.0.0.
...

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

N. J. A. Sloane, Maple programs for A211097 etc.

Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof), A211100.

Cf. A211095-A211099.

A329355 The binary expansion of a(n) is the second through n-th terms of A000002 - 1.

Original entry on oeis.org

0, 1, 3, 6, 12, 25, 50, 101, 203, 406, 813, 1627, 3254, 6508, 13017, 26034, 52068, 104137, 208275, 416550, 833101, 1666202, 3332404, 6664809, 13329618, 26659237, 53318475, 106636950, 213273900, 426547801, 853095602, 1706191204, 3412382409, 6824764818

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

			a(11) = 813 has binary expansion q = {1, 1, 0, 0, 1, 0, 1, 1, 0, 1}, and q + 1 is {2, 2, 1, 1, 2, 1, 2, 2, 1, 2}, which is the second through 11th terms of A000002.

Replacing "A000002 - 1" with "2 - A000002" gives A329356.
Partial sums of A000002 are A054353.
Initial subsequences of A000002 are A329360.
Cf. A211100, A275692, A288605, A296658, A329315, A329316, A329317, A329327, A329361, A329362.

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=If[n==0,{},Nest[kolagrow,{1},n-1]];
Table[FromDigits[kol[n]-1,2],{n,30}]

A329356 The binary expansion of a(n) is the first n terms of 2 - A000002.

Original entry on oeis.org

0, 1, 2, 4, 9, 19, 38, 77, 154, 308, 617, 1234, 2468, 4937, 9875, 19750, 39501, 79003, 158006, 316012, 632025, 1264050, 2528101, 5056203, 10112406, 20224813, 40449626, 80899252, 161798505, 323597011, 647194022, 1294388045, 2588776091, 5177552182, 10355104365

Offset: 0

Gus Wiseman, Nov 12 2019

			a(7) = 77 has binary expansion q = {1, 0, 0, 1, 1, 0, 1}, and 2 - q is {1, 2, 2, 1, 1, 2, 1}, which is the first 7 terms of A000002.

Cf. A118270, A329361.
Replacing "2 - A000002" with "A000002 - 1" gives A329355.
Initial subsequences of A000002 are A329360.
Cf. A121016, A211100, A275692, A296658, A329315, A329316, A329317, A329362.

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=If[n==0,{},Nest[kolagrow,{1},n-1]];
Table[FromDigits[2-kol[n],2],{n,0,30}]

a(n) = floor((1-c/2)*2^n), where c = A118270 is the Kolakoski constant. - Lorenzo Sauras Altuzarra, Jan 01 2023

cp's OEIS Frontend

A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

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A211096 Smallest (i.e., rightmost) Lyndon word in the Lyndon factorization of the binary representation of n (written using 1's and 2's rather than 0's and 1's, since numbers > 0 in the OEIS cannot begin with 0).

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A334297 Length of the Lyndon factorization of the reversed n-th composition in standard order.

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A329396 Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.

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A329399 Numbers whose reversed binary expansion has uniform Lyndon factorization.

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A329400 Length of the co-Lyndon factorization of the binary expansion of n with the most significant (first) digit removed.

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A329401 Numbers whose binary expansion without the most significant (first) digit is a co-Lyndon word.

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A211098 Length of largest (i.e., leftmost) Lyndon word in Lyndon factorization of binary vectors of lengths 1, 2, 3, ...

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A329355 The binary expansion of a(n) is the second through n-th terms of A000002 - 1.

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