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 1-10 of 15 results. Next

A378286 a(n) = number of terms in row n of the array A378282.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 11, 16, 23, 35, 51, 75, 113, 168, 249, 374, 560, 840, 1262, 1896, 2844, 4270, 6405, 9607, 14421, 21633, 32449, 48676, 73027, 109546, 164334, 246528, 369784, 554686, 832046, 1248073, 1872168, 2808326, 4212710, 6319089, 9478842
Offset: 1

Views

Author

Clark Kimberling, Dec 20 2024

Keywords

Examples

			First eleven rows of array A378282:
1
1
2
1  1
1  2
2  1  1
1  1  2  1
1  2  1  1  2
2  1  1  2  1  2  2
1  1  2  1  2  2  1  2  2  1  1
1  2  1  1  2  1  1  2  2  1  2  2  1  1  2  2  1
a(11) = 16 = number of terms in row 11.

Crossrefs

Cf. A378282.

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}; rows = {row1};
col = PadRight[{}, 30, {1, 1, 2}](*112112112112112112...*);
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm; (* A378282 *)
Map[Length, rows]  (* this sequence *)
(* Peter J. C. Moses, Nov 21 2024 *)

A378303 Irregular triangular T: (row 1) = (1); (row n+1) = inverse runlength sequence of row n, starting with 1 if r = 3k for some k, and 2 otherwise. See Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 20 2024

Keywords

Comments

See A378282 for a guide to related arrays and sequences.

Examples

			First nine rows:
   1
   2
   2, 2
   1, 1, 2, 2
   2, 1, 2, 2, 1, 1
   2, 2, 1, 2, 2, 1, 1, 2, 1
   1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1
   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, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2
(row 9) = (2,2,1,2,2,1,...) has runlength sequence (2,1,2,2,...) = (row 8).

Crossrefs

Cf. A378282, A378304, A378305, A378306, A378307 (row lengths)

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}; rows = {row1};
col = PadRight[{}, 18, {1, 2, 2}]
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
Flatten[rows]  (* sequence *)
(* Peter J. C. Moses, Nov 21 2024 *)

A378283 Unique sequence s starting with 1,1,2,1 such that if r(r(r(s) = s and r(s) != s and r(r(s) != s, where r(#) denotes the runlength sequence of a sequence #.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 16 2024

Keywords

Comments

This sequence is one of three limiting rows of the array A378282. See A378282 for a guide to related arrays and sequences.

Crossrefs

Cf. A378282, A378284, A378285, A378286, A378303.

Programs

  • Mathematica
    z = 18;
invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = {1}; rows = {row1};
col = PadRight[{}, z, {1, 1, 2}]
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm
Flatten[rows]   (* A378282 *)
rows[[z - 2]];  (* A378283 *)
rows[[z - 1]];  (* A378284 *)
rows[[z]];      (* A378285 *)
Map[Length, rows]   (* A378286 *)
(* Peter J. C. Moses, Nov 21 2024 *)

A378284 Unique sequence s starting with 1,2,1 such that if r(r(r(s) = s and r(s) != s and r(r(s) != s, where r(#) denotes the runlength sequence of a sequence #.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 20 2024

Keywords

Comments

This sequence is one of three limiting rows of the array A378282. See A378282 for a guide to related arrays and sequences.

Crossrefs

Cf. A378282, A378283, A378285, A378286, A378303.

Programs

  • Mathematica
    z = 18;
invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = {1}; rows = {row1};
col = PadRight[{}, z, {1, 1, 2}]
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm
Flatten[rows]   (* A378282 *)
rows[[z - 2]];  (* A378283 *)
rows[[z - 1]];  (* A378284 *)
rows[[z]];      (* A378285 *)
Map[Length, rows]   (* A378286 *)
(* Peter J. C. Moses, Nov 21 2024 *)

A378285 Unique sequence s starting with 2,1 such that if r(r(r(s))) = s and r(s) != s and r(r(s)) != s, where r(#) denotes the runlength sequence of a sequence #.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 20 2024

Keywords

Comments

This sequence is one of three limiting rows of the array A378282. See A378282 for a guide to related arrays and sequences.

Crossrefs

Cf. A378282, A378283, A378284, A378286, A378303.

Programs

  • Mathematica
    z = 18;
invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = {1}; rows = {row1};
col = PadRight[{}, z, {1, 1, 2}]
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm
Flatten[rows]   (* A378282 *)
rows[[z - 2]];  (* A378283 *)
rows[[z - 1]];  (* A378284 *)
rows[[z]];      (* A378285 *)
Map[Length, rows]   (* A378286 *)
(* Peter J. C. Moses, Nov 21 2024 *)

A378304 Unique sequence s starting with 1,1,2,2 such that if r(r(r(s))) = s and r(s) != s and r(r(s)) != s, where r(#) denotes the runlength sequence of a sequence #.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 20 2024

Keywords

Comments

This sequence is one of three limiting rows of the array A378303. See A378282 for a guide to related arrays and sequences.

Crossrefs

Cf. A378282, A378303, A378305, A378306, A378307.

Programs

  • Mathematica
    z = 18;
invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = {1}; rows = {row1};
col = PadRight[{}, z, {1, 2, 2}]
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm  (* A378303 *)
rows[[z - 2]];      (* A378304 *)
rows[[z - 1]];      (* A378305 *)
rows[[z]];          (* A378306 *)
Map[Length, rows] (* A378307 *)
(* Peter J. C. Moses, Nov 21 2024 *)

A378305 Unique sequence s starting with 2,1,2 such that if r(r(r(s))) = s and r(s) != s and r(r(s)) != s, where r(#) denotes the runlength sequence of a sequence #.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 20 2024

Keywords

Comments

This sequence is one of three limiting rows of the array A378303. See A378282 for a guide to related arrays and sequences.

Crossrefs

Cf. A378282, A378303, A378304, A378306, A378307.

Programs

  • Mathematica
    z = 18;
invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = {1}; rows = {row1};
col = PadRight[{}, z, {1, 2, 2}]
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm  (* A378303 *)
rows[[z - 2]];      (* A378304 *)
rows[[z - 1]];      (* A378305 *)
rows[[z]];          (* A378306 *)
Map[Length, rows] (* A378307 *)
(* Peter J. C. Moses, Nov 21 2024 *)

A378306 Unique sequence s starting with 2,2 such that if r(r(r(s))) = s and r(s) != s and r(r(s)) != s, where r(#) denotes the runlength sequence of a sequence #.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 20 2024

Keywords

Comments

This sequence is one of three limiting rows of the array A378303. See A378282 for a guide to related arrays and sequences.

Crossrefs

Cf. A378282, A378303, A378304, A378305, A378307.

Programs

  • Mathematica
    z = 18;
invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = {1}; rows = {row1};
col = PadRight[{}, z, {1, 2, 2}]
Do[AppendTo[rows, invRE[Last[rows], col[[n]]]], {n, 2, Length[col]}]
rows // ColumnForm  (* A378303 *)
rows[[z - 2]];      (* A378304 *)
rows[[z - 1]];      (* A378305 *)
rows[[z]];          (* A378306 *)
Map[Length, rows] (* A378307 *)
(* Peter J. C. Moses, Nov 21 2024 *)

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 *)

A378397 Rectangular array read by descending antidiagonals: (row 1) = u, and for n >= 2, (row n) = u-inverse runlength sequence of u, where u = A003842 (an infinite Fibonacci word). See Comments.

Original entry on oeis.org

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

Crossrefs

Cf. A003842, A378282, A378396, 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 = SubstitutionSystem[{1 -> {1, 2}, 2 -> {1}}, {1}, {7}][[1]] (* A003842 *);
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 *)
Showing 1-10 of 15 results. Next

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