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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.

%I A378396 #9 Jan 11 2025 04:10:09
%S A378396 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,
%T A378396 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,
%U A378396 2,2,1,2,1,1,2,2,2,2
%N 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.
%C A378396 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.
%e A378396 The corner of the array begins:
%e A378396    1  2  2  1  2  1  1  2  2  1  1  2  1  2  2  1  2  1  1  2  1
%e A378396    2  1  1  2  2  1  2  2  1  2  1  1  2  2  1  2  1  1  2  1  1
%e A378396    2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1
%e A378396    1  1  2  2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  2
%e A378396    2  1  2  2  1  1  2  1  1  2  2  1  2  2  1  2  1  1  2  2  1
%e A378396    1  1  2  1  1  2  2  1  2  1  1  2  1  2  2  1  1  2  1  1  2
%e A378396    1  2  1  1  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2  2  1
%e A378396    2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2
%e A378396    2  2  1  2  1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1
%e A378396    1  1  2  2  1  2  2  1  2  1  1  2  1  2  2  1  1  2  1  1  2
%e A378396    1  2  1  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2  2
%e A378396    2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1  2
%e A378396    ...
%t A378396 invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
%t A378396     Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
%t A378396 row1 = 1 + ThueMorse[Range[0, 20]]   (* 1 + A010060 *);
%t A378396 rows = {row1}; col = Take[row1, 12];
%t A378396 Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
%t A378396 rows // ColumnForm  (* array *)
%t A378396 w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
%t A378396 (* _Peter J. C. Moses_, Nov 20 2024 *)
%Y A378396 Cf. A010060, A378282, A378397, A378398, A378399, A378401.
%K A378396 nonn,tabl
%O A378396 1,2
%A A378396 _Clark Kimberling_, Dec 21 2024