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 1-5 of 5 results.

A378396 Rectangular array read by descending antidiagonals: (row 1) = u, and for n >= 2, (row n) = u-inverse runlength sequence of u, where u = 1 + A010060. See Comments.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2
Offset: 1

Views

Author

Clark Kimberling, Dec 21 2024

Keywords

Comments

If u and v are sequences, both consisting of 1's and 2's, we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array, in which each row after the first is an inverse runlength sequence of the preceding row, is determined by its first column. Generally, if the first column is periodic with fundamental period p, then the array has p distinct limiting sequences; otherwise, there is no limiting sequence; however, if a segment, of any length, occurs in a row, then it also occurs in a subsequent row. See A378282 for details and related sequences.

Examples

			The corner of the array begins:
   1  2  2  1  2  1  1  2  2  1  1  2  1  2  2  1  2  1  1  2  1
   2  1  1  2  2  1  2  2  1  2  1  1  2  2  1  2  1  1  2  1  1
   2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1
   1  1  2  2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  2
   2  1  2  2  1  1  2  1  1  2  2  1  2  2  1  2  1  1  2  2  1
   1  1  2  1  1  2  2  1  2  1  1  2  1  2  2  1  1  2  1  1  2
   1  2  1  1  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2  2  1
   2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2
   2  2  1  2  1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1
   1  1  2  2  1  2  2  1  2  1  1  2  1  2  2  1  1  2  1  1  2
   1  2  1  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2  2
   2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1  2
   ...

Crossrefs

Cf. A010060, A378282, A378397, A378398, A378399, A378401.

Programs

  • Mathematica
    invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = 1 + ThueMorse[Range[0, 20]]   (* 1 + A010060 *);
rows = {row1}; col = Take[row1, 12];
Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Nov 20 2024 *)

A378398 Rectangular array read by descending antidiagonals: (row 1) = u, and for n >= 2, (row n) = u-inverse runlength sequence of u, where u = A014675 (an infinite Fibonacci word). See Comments.

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2
Offset: 1

Views

Author

Clark Kimberling, Dec 21 2024

Keywords

Comments

If u and v are sequences, both consisting of 1's and 2's, we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array, in which each row after the first is an inverse runlength sequence of the preceding row, is determined by its first column. Generally, if the first column is periodic with fundamental period p, then the array has p distinct limiting sequences; otherwise, there is no limiting sequence; however, if a segment, of any length, occurs in a row, then it also occurs in a subsequent row. See A378282 for details and related sequences.

Examples

			The corner of the array begins:
     2  1  2  2  1  2  1  2  2  1  2  2  1  2  1  2  2  1  2  1  2
     1  1  2  1  1  2  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1
     2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  1  2  2  1
     2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  1  2  1  1  2  2
     1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2  1  1  2
     2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1  1  2  2  1
     1  1  2  1  1  2  2  1  2  1  1  2  1  2  2  1  1  2  1  1  2
     2  1  2  2  1  2  1  1  2  2  1  2  2  1  2  1  1  2  1  1  2
     2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  1  2  1  1  2  2
     1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2  1  1  2
     2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1  1  2  2  1
     2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2  2  1  2  2  1

Crossrefs

Cf. A014675, A378282, A378396, A378397, A378399, A378401.

Programs

  • Mathematica
    invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = SubstitutionSystem[{1 -> {2}, 2 -> {2, 1}}, {1}, {7}][[1]] (* A014675 *);
rows = {row1}; col = Take[row1, 12];
Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Nov 20 2024 *)

A378399 Rectangular array read by descending antidiagonals: (row 1) = u, and for n >= 2, (row n) = u-inverse runlength sequence of u, where u = A006337 (a Beatty difference sequence). See Comments.

Original entry on oeis.org

1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Clark Kimberling, Dec 21 2024

Keywords

Comments

If u and v are sequences, both consisting of 1's and 2's, we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array, in which each row after the first is an inverse runlength sequence of the preceding row, is determined by its first column. Generally, if the first column is periodic with fundamental period p, then the array has p distinct limiting sequences; otherwise, there is no limiting sequence; however, if a segment, of any length, occurs in a row, then it also occurs in a subsequent row. See A378282 for details and related sequences.

Examples

			The corner of the array begins:
   1  2  1  2  1  1  2  1  2  1  1  2  1  2  1  2  1  1  2
   2  1  1  2  1  1  2  1  2  2  1  2  2  1  2  1  1  2  1
   1  1  2  1  2  2  1  2  1  1  2  1  1  2  2  1  2  2  1
   2  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2  1
   1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
   1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1
   2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
   1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1
   2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2
   1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
   1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1
   2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
   1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1

Crossrefs

Cf. A006337, A378282, A378396, A378397, A378398, A378401.

Programs

  • Mathematica
    invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = Differences[Table[Floor[n*Sqrt[2]], {n, 1, 20}]] (*  A006337 );
rows = {row1}; col = Take[row1, 12];
Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Nov 20 2024 *)

A378401 Rectangular array read by descending antidiagonals: (row 1) = u, and for n >= 2, (row n) = u-inverse runlength sequence of u, where u = A006338. See Comments.

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Dec 21 2024

Keywords

Comments

If u and v are sequences, both consisting of 1's and 2's, we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array, in which each row after the first is an inverse runlength sequence of the preceding row, is determined by its first column. Generally, if the first column is periodic with fundamental period p, then the array has p distinct limiting sequences; otherwise, there is no limiting sequence; however, if a segment, of any length, occurs in a row, then it also occurs in a subsequent row. See A378282 for details and related sequences.

Examples

			The corner of the array begins:
     2  1  2  1  1  2  1  2  1  2  1  1  2  1  2  1  1  2  1
     1  1  2  1  1  2  1  2  2  1  2  2  1  2  2  1  2  1  1
     2  1  2  2  1  2  1  1  2  1  1  2  2  1  2  2  1  1  2
     1  1  2  1  1  2  2  1  2  2  1  2  1  1  2  1  2  2  1
     1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  2  1
     2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
     1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1
     2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2
     1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
     2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2
     1  1  2  1  1  2  2  1  2  2  1  2  1  1  2  2  1  2  2
     1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  2  1

Crossrefs

Cf. A006338, A378282, A378396, A378397, A378398, A378399.

Programs

  • Mathematica
    invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = Differences[Table[Floor[n*Sqrt[2]+1/2], {n, 1, 20}]];
rows = {row1}; col = Take[row1, 12];
Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Nov 20 2024 *)

A380560 Rectangular array R, read by descending antidiagonals: (row 1) = (R(1,k)) = (A006337(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = 1 for n >=1, See Comments.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2025

Keywords

Comments

For present purposes, all sequences to be considered consist entirely of 1s and 2s. If u and v are such sequences (infinite or finite), we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array (in which each row after the first is an inverse runlength sequence of the preceding row) is determined by its first column. In this array, the first column consists solely of 1s. No two rows are identical.
Row 1: A006337; limiting row: A000002 (Kolakoski sequence).
Guide to related sequences (arrays):
A378282, limiting sequences of periodic inverse runlength arrays
A380560, (row 1)=A006337, periodic (column 1)=(1,...)
A380561, (row 1)=A006337, periodic (column 1)=(1,1,2,...)
A380562, (row 1)=A006337, periodic (column 1)=(1,2,2,...)
A380563, (row 1)=1+A010060, periodic (column 1)=(1,...)
A378303, (ro1 1)=A006337, periodic (column 1)=(2,2,1,...)
A378396, self-inverse runlength array, u = v = 1+A010060
A378397, self-inverse runlength array, u = v = A003842
A378398, self-inverse runlength array, u = v = A014675
A378399, self-inverse runlength array, u = v = A006337

Examples

			Corner:
  1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2 1 1 2 1
  1 2 2 1 2 2 1 2 1 1 2 1 1 2 1 2 2 1 2 2
  1 2 2 1 1 2 1 1 2 2 1 2 2 1 2 1 1 2 1 2
  1 2 2 1 1 2 1 2 2 1 2 1 1 2 2 1 2 2 1 1
  1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 1 2
  1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1
  1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1

Crossrefs

Cf. A000002, A000012 (column 1), A006337.

Programs

  • Mathematica
    invR[seq_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {1, 2}, {2, -1, 2}], 2]]];
s = Differences[Table[Floor[n*Sqrt[2]], {n, 1, 21}]]; (* A006337 *)
t = NestList[invR, s, 12];
u[n_] := Take[t[[n]], 20];
Table[u[n], {n, 1, 12}]  (* array *)
v[n_, k_] := t[[n]][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Nov 13 2024 *)
Showing 1-5 of 5 results.

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