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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.

%I A378399 #8 Jan 11 2025 04:10:41
%S A378399 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,
%T A378399 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,
%U A378399 1,1,1,2,2,2,2,2,2,2
%N 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.
%C A378399 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 A378399 The corner of the array begins:
%e A378399    1  2  1  2  1  1  2  1  2  1  1  2  1  2  1  2  1  1  2
%e A378399    2  1  1  2  1  1  2  1  2  2  1  2  2  1  2  1  1  2  1
%e A378399    1  1  2  1  2  2  1  2  1  1  2  1  1  2  2  1  2  2  1
%e A378399    2  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2  1
%e A378399    1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
%e A378399    1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1
%e A378399    2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
%e A378399    1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1
%e A378399    2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2
%e A378399    1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
%e A378399    1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1
%e A378399    2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
%e A378399    1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1
%t A378399 invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
%t A378399     Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
%t A378399 row1 = Differences[Table[Floor[n*Sqrt[2]], {n, 1, 20}]] (*  A006337 );
%t A378399 rows = {row1}; col = Take[row1, 12];
%t A378399 Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
%t A378399 rows // ColumnForm  (* array *)
%t A378399 w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
%t A378399 (* _Peter J. C. Moses_, Nov 20 2024 *)
%Y A378399 Cf. A006337, A378282, A378396, A378397, A378398, A378401.
%K A378399 nonn,tabl
%O A378399 1,2
%A A378399 _Clark Kimberling_, Dec 21 2024