cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Rémy Sigrist, Oct 01 2024

Keywords

Comments

Every positive integer appears twice in this sequence.

Examples

			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

Links

Crossrefs

Cf. A351653, A376637, A376698, A376733.

Programs

  • PARI
    \\ See Links section.

Formula

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

A376698 a(n) is the least k >= 0 such that the k-th iterate of the run lengths transform of A376637(n) equals 1.

Original entry on oeis.org

0, 1, 2, 3, 3, 2, 4, 4, 4, 4, 5, 3, 5, 5, 3, 5, 5, 6, 5, 6, 6, 5, 6, 5, 4, 4, 4, 4, 6, 7, 6, 6, 6, 7, 7, 6, 6, 6, 7, 6, 7, 6, 6, 7, 7, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 7, 8, 7, 8, 7, 7, 7, 7, 8, 7, 8, 7, 7, 7, 8, 7, 8, 7, 8, 7, 7, 8, 8, 7, 7, 8, 7, 8, 7, 8, 7
Offset: 1

Views

Author

Rémy Sigrist, Oct 02 2024

Keywords

Comments

For any k > 0, the value k appears 2^(k-1) times.

Examples

			The first terms, alongside the corresponding run lengths transforms, are:
  n   a(n)  Run lengths transforms
  --  ----  ---------------------------------
   1     0  1
   2     1  2 -> 1
   3     2  11 -> 2 -> 1
   4     3  12 -> 11 -> 2 -> 1
   5     3  21 -> 11 -> 2 -> 1
   6     2  22 -> 2 -> 1
   7     4  112 -> 21 -> 11 -> 2 -> 1
   8     4  122 -> 12 -> 11 -> 2 -> 1
   9     4  211 -> 12 -> 11 -> 2 -> 1
  10     4  221 -> 21 -> 11 -> 2 -> 1
  11     5  1121 -> 211 -> 12 -> 11 -> 2 -> 1
  12     3  1122 -> 22 -> 2 -> 1
  13     5  1211 -> 112 -> 21 -> 11 -> 2 -> 1
  14     5  2122 -> 112 -> 21 -> 11 -> 2 -> 1
  15     3  2211 -> 22 -> 2 -> 1
  16     5  2212 -> 211 -> 12 -> 11 -> 2 -> 1

Links

Crossrefs

Cf. A376637, A376676.

Programs

  • PARI
    \\ See Links section.

Formula

a(n) = a(A376676(n)) + 1 for any n > 1.

A376685 a(n) is the unique k such that A376637(k) is the reversal of A376637(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 9, 10, 7, 8, 13, 15, 11, 16, 12, 14, 19, 21, 17, 23, 18, 24, 20, 22, 27, 28, 25, 26, 38, 34, 40, 33, 32, 30, 39, 37, 36, 29, 35, 31, 44, 43, 42, 41, 48, 47, 46, 45, 54, 53, 56, 55, 50, 49, 52, 51, 59, 60, 57, 58, 62, 61, 64, 63, 67, 68, 65
Offset: 1

Views

Author

Rémy Sigrist, Oct 01 2024

Keywords

Comments

This sequence is a self-inverse permutation of the positive integers.

Examples

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

Links

Crossrefs

Cf. A004086, A376637, A376686.

Programs

  • PARI
    \\ See Links section.

Formula

A376637(a(n)) = A004086(A376637(n)).

A376638 a(n) is the number of n-digit numbers in A376637.

Original entry on oeis.org

2, 4, 4, 6, 8, 4, 12, 8, 8, 12, 20, 0, 16, 16, 12, 24, 16, 8, 12, 16, 32, 12, 20, 32, 24, 16, 16, 12, 16, 20, 40, 40, 8, 28, 44, 20, 44, 28, 20, 24, 20, 20, 16, 32, 20, 44, 44, 56, 28, 28, 20, 60, 52, 24, 56, 56, 20, 36, 36, 32, 24, 24, 32, 24, 16, 60, 16, 32
Offset: 1

Views

Author

Rémy Sigrist, Sep 30 2024

Keywords

Comments

All terms are even: if a word w belongs to A376637, then replacing 1's by 2's and 2's by 1's in w yields another word in A376637.
a(n)/2 is the number of terms of A376637 with digital sum n.

Examples

			Sequence A376637 begins: 1, 2,  11, 12, 21, 22,  112, 122, 211, 221,  1121.
So a(1) = 2, a(2) = 4, a(3) = 4.

Links

Crossrefs

Cf. A007782, A376637.

A376674 a(n) is the unique number k different from n such that the words A376637(n) and A376637(k) have the same run lengths transform.

Original entry on oeis.org

2, 1, 6, 5, 4, 3, 10, 9, 8, 7, 16, 15, 14, 13, 12, 11, 24, 23, 22, 21, 20, 19, 18, 17, 28, 27, 26, 25, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 48, 47, 46, 45, 44, 43, 42, 41, 56, 55, 54, 53, 52, 51, 50, 49, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58
Offset: 1

Views

Author

Rémy Sigrist, Oct 01 2024

Keywords

Comments

Also, replacing 1's by 2's and 2's by 1's in A376637(n) yields A376637(a(n)).
This sequence is a self-inverse permutation of the positive integers without fixed points.

Examples

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

Links

Crossrefs

Cf. A351653, A376637.

Programs

  • PARI
    \\ See Links section.

Formula

A351653(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) = 2*a(A376676(n)) otherwise a(n) = 2*a(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

Views

Author

Rémy Sigrist, Oct 03 2024

Keywords

Comments

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.

Examples

			See illustration in Links section.

Links

Crossrefs

Cf. A070939, A376637, A376676, A376698.

Programs

  • PARI
    \\ See Links section.

Formula

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

A376677 List of subwords (or factors) of the Kolakoski sequence (A000002).

Original entry on oeis.org

1, 2, 11, 12, 21, 22, 112, 121, 122, 211, 212, 221, 1121, 1122, 1211, 1212, 1221, 2112, 2121, 2122, 2211, 2212, 11211, 11212, 11221, 12112, 12122, 12211, 12212, 21121, 21122, 21211, 21221, 22112, 22121, 22122, 112112, 112122, 112212, 121121, 121122, 121221
Offset: 1

Views

Author

Rémy Sigrist, Oct 01 2024

Keywords

Comments

There are A007782(m) terms with m digits.

Links

Crossrefs

Cf. A000002, A007782, A214215, A376637.

Programs

  • PARI
    \\ See Links section.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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