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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.

%I A378397 #8 Jan 11 2025 04:10:23
%S A378397 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,
%T A378397 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,
%U A378397 1,2,1,1,2,1,1,1,1,1
%N 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.
%C A378397 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 A378397 The corner of the array begins:
%e A378397    1  2  1  1  2  1  2  1  1  2  1  1  2  1  2  1  1  2  1  2  1
%e A378397    2  1  1  2  1  2  2  1  2  2  1  2  1  1  2  1  2  2  1  2  2
%e A378397    1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  2  1  2
%e A378397    1  2  1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
%e A378397    2  1  1  2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  2
%e A378397    1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  1  2
%e A378397    2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1  2
%e A378397    1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2
%e A378397    1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1
%e A378397    2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2
%e A378397    1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2
%e A378397    1  2  1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
%t A378397 invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
%t A378397     Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
%t A378397 row1 = SubstitutionSystem[{1 -> {1, 2}, 2 -> {1}}, {1}, {7}][[1]] (* A003842 *);
%t A378397 rows = {row1}; col = Take[row1, 12];
%t A378397 Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
%t A378397 rows // ColumnForm  (* array *)
%t A378397 w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
%t A378397 (* _Peter J. C. Moses_, Nov 20 2024 *)
%Y A378397 Cf. A003842, A378282, A378396, A378398, A378399, A378401.
%K A378397 nonn,tabl
%O A378397 1,2
%A A378397 _Clark Kimberling_, Dec 21 2024