A380563 Rectangular array R read by descending antidiagonals: (row 1) = (R(1,k)) = (1 + A010060(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = (1, 1, 1, 1, 1,...) = (A130196(n)) for n >= 1. See Comments.

1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1

Offset

1,2

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 is the periodic sequence with period 1. The limiting sequence is A000002 (Kolakoski sequence). No two rows are identical.

Row 1: 1 + A010060; i.e., R(n,1) = 1 + n-th term op the Thue-Morse sequence.

See A380560 for a guide to related sequences.

Links

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

Example

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

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, 200]] (* 1 + A010060 *);
rows = {row1};
col = PadRight[{}, 30, {1}];
Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
v[n_, k_] := rows[[n]][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*sequence*)
(*_Peter J.C.Moses_, Nov 20 2024*)

Crossrefs

Cf. A000002, A000012 (column 1), A010060.

Sequence in context: A161175 A356515 A095955 * A272772 A293431 A078573

Adjacent sequences: A380560 A380561 A380562 * A380564 A380565 A380566

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 27 2025

Status

approved