A376698 -id:A376698 - cp's OEIS frontend

A376637 The word 1 belongs to the sequence, and whenever a word w belongs to the sequence, then the words consisting of 1's and 2's whose run lengths transform equals w also belong to the sequence.

1, 2, 11, 12, 21, 22, 112, 122, 211, 221, 1121, 1122, 1211, 2122, 2211, 2212, 11221, 12112, 12211, 12212, 21121, 21122, 21221, 22112, 112212, 121122, 212211, 221121, 1121122, 1121221, 1122122, 1211221, 1221121, 1221211, 2112122, 2112212, 2122112, 2211211

Offset: 1

Rémy Sigrist, Sep 30 2024

This sequence lists finite smooth words: finite words w composed of 1's and 2's without three or more consecutive equal digits, such that for any k > 0, the k-th iterate of the run lengths transform of w is also a word composed of 1's and 2's without three or more consecutive equal digits.

			The first terms, alongside their run lengths transform, are:
  n   a(n)  RL(a(n))
  --  ----  --------
   1     1         1
   2     2         1
   3    11         2
   4    12        11
   5    21        11
   6    22         2
   7   112        21
   8   122        12
   9   211        12
  10   221        21
  11  1121       211
  12  1122        22
  13  1211       112
  14  2122       112
  15  2211        22
  16  2212       211

Geneviève Paquin, Srĕcko Brlek, Damien Jamet, Extremal generalized smooth words
Rémy Sigrist, Table of n, a(n) for n = 1..10048
Rémy Sigrist, Illustration of the first terms (arrows denotes run lengths transforms)
Rémy Sigrist, PARI program
Index entries for sequences related to Kolakoski sequence

Cf. A000002, A007931, A376638, A376674, A376676, A376685, A376698.

PARI
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A376676 a(n) is the unique k such that the run lengths transform of A376637(n) equals A376637(k).

Original entry on oeis.org

1, 1, 2, 3, 3, 2, 5, 4, 4, 5, 9, 6, 7, 7, 6, 9, 10, 11, 8, 13, 13, 8, 11, 10, 15, 12, 12, 15, 14, 21, 16, 17, 19, 18, 18, 19, 17, 16, 21, 14, 23, 22, 24, 20, 20, 24, 22, 23, 27, 28, 25, 26, 26, 25, 28, 27, 38, 30, 29, 34, 33, 32, 32, 33, 34, 29, 30, 38, 37, 36

Offset: 1

Rémy Sigrist, Oct 01 2024

Every positive integer appears twice in this sequence.

			The first terms, alongside the corresponding terms of A376637, are:
  n   a(n)  A376637(n)  A376637(a(n))
  --  ----  ----------  -------------
   1     1           1              1
   2     1           2              1
   3     2          11              2
   4     3          12             11
   5     3          21             11
   6     2          22              2
   7     5         112             21
   8     4         122             12
   9     4         211             12
  10     5         221             21
  11     9        1121            211
  12     6        1122             22
  13     7        1211            112
  14     7        2122            112
  15     6        2211             22
  16     9        2212            211

Rémy Sigrist, Table of n, a(n) for n = 1..10048
Rémy Sigrist, PARI program

Cf. A351653, A376637, A376698, A376733.

PARI
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A376637(a(n)) = A351653(A376637(n)).

A376733 a(1) = 0; for any any n > 1, if A376637(n) starts with a digit 1 then a(n) = 2a(A376676(n)) otherwise a(n) = 2a(A376676(n)) + 1.

Original entry on oeis.org

0, 1, 2, 4, 5, 3, 10, 8, 9, 11, 18, 6, 20, 21, 7, 19, 22, 36, 16, 40, 41, 17, 37, 23, 14, 12, 13, 15, 42, 82, 38, 44, 32, 72, 73, 33, 45, 39, 83, 43, 74, 34, 46, 80, 81, 47, 35, 75, 26, 30, 28, 24, 25, 29, 31, 27, 78, 164, 84, 144, 64, 88, 89, 65, 145, 85, 165

Offset: 1

Rémy Sigrist, Oct 03 2024

The binary expansion of a(n) encodes the position of A376637(n) within the binary tree underlying A376676 (see illustration in Links section).

This sequence is a bijection from the positive integers to the nonnegative integers.

			See illustration in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10048
Rémy Sigrist, Illustration of terms < 32 (where b = A376637)
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A070939, A376637, A376676, A376698.

PARI
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A070939(a(n)) = A376698(n) for any n > 0.

cp's OEIS Frontend

A376637 The word 1 belongs to the sequence, and whenever a word w belongs to the sequence, then the words consisting of 1's and 2's whose run lengths transform equals w also belong to the sequence.

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PARI

A376676 a(n) is the unique k such that the run lengths transform of A376637(n) equals A376637(k).

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PARI

Formula

A376733 a(1) = 0; for any any n > 1, if A376637(n) starts with a digit 1 then a(n) = 2a(A376676(n)) otherwise a(n) = 2a(A376676(n)) + 1.

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Author

Keywords

Comments

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PARI

Formula

cp's OEIS Frontend

A376637 The word 1 belongs to the sequence, and whenever a word w belongs to the sequence, then the words consisting of 1's and 2's whose run lengths transform equals w also belong to the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A376676 a(n) is the unique k such that the run lengths transform of A376637(n) equals A376637(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A376733 a(1) = 0; for any any n > 1, if A376637(n) starts with a digit 1 then a(n) = 2*a(A376676(n)) otherwise a(n) = 2*a(A376676(n)) + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A376733 a(1) = 0; for any any n > 1, if A376637(n) starts with a digit 1 then a(n) = 2a(A376676(n)) otherwise a(n) = 2a(A376676(n)) + 1.