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A380562 Rectangular array R read by descending antidiagonals: (row 1) = (R(1,k)) = (A006337(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = (1, 2, 2, 1, 2, 2, 1, 2, 2, ...) = (A130196(n)) for n >= 1. See Comments.

%I A380562 #9 Jun 11 2025 03:04:30
%S A380562 1,2,2,1,1,2,2,1,2,1,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,2,2,1,1,2,1,1,2,1,
%T A380562 1,2,2,1,2,2,1,2,2,1,2,1,2,1,2,1,2,2,2,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,
%U A380562 2,2,1,1,2,1,2,2,1,2
%N A380562 Rectangular array R read by descending antidiagonals: (row 1) = (R(1,k)) = (A006337(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = (1, 2, 2, 1, 2, 2, 1, 2, 2, ...) = (A130196(n)) for n >= 1. See Comments.
%C A380562 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 1,1,2. There are three limiting sequences: A378283, A378284, A378285. No two rows are identical.
%C A380562 See A380560 for a guide to related sequences.
%e A380562 Corner:
%e A380562     1  2  1  2  1  1  2  1  2  1  1  2  1  2  1  2  1  1  2
%e A380562     2  1  1  2  1  1  2  1  2  2  1  2  2  1  2  1  1  2  1
%e A380562     2  2  1  2  1  1  2  1  2  2  1  2  2  1  1  2  1  1  2
%e A380562     1  1  2  2  1  2  2  1  2  1  1  2  1  1  2  2  1  2  2
%e A380562     2  1  2  2  1  1  2  1  1  2  2  1  2  2  1  2  1  1  2
%e A380562     2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2  2  1  2
%e A380562     1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  2  1  1  2
%e A380562     2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1  1  2
%e A380562     2  2  1  2  2  1  1  2  1  2  2  1  2  1  1  2  2  1  2
%e A380562     1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  2  1  1  2
%t A380562 invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
%t A380562     Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
%t A380562 row1 = Flatten[Table[Nest[Flatten[# /. {1 -> {1, 2}, 2 -> {1, 1, 2}}] &, {1}, n], {n, 3}]]; (* A006337 *)
%t A380562 rows = {row1};
%t A380562 col = PadRight[{}, 30, {1, 2, 2}];
%t A380562 Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
%t A380562 rows // ColumnForm  (* array *)
%t A380562 v[n_, k_] := rows[[n]][[k]];
%t A380562 Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*sequence*)
%t A380562 (* _Peter J. C. Moses_, Nov 20 2024 *)
%Y A380562 Cf. A000002, A130196 (column 1), A006337, A380560.
%K A380562 nonn,tabl
%O A380562 1,2
%A A380562 _Clark Kimberling_, Jan 27 2025