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.

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

1,2

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.

Links

Table of n, a(n) for n=1..78.

Example

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
   ...

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 *)

Crossrefs

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

Sequence in context: A191516 A168141 A232654 * A105971 A375583 A375582

Adjacent sequences: A378393 A378394 A378395 * A378397 A378398 A378399

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 21 2024

Status

approved