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.

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

Author

Clark Kimberling, Dec 21 2024

Keywords

• nonn
• tabl

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 *)