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A380561 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, 1, 2, 1, 1, 2, 1, 1, 2, ...) = (A100063(n)) for n >= 1. See Comments.

%I A380561 #13 Jun 11 2025 03:03:07
%S A380561 1,2,1,1,2,2,2,2,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,2,1,1,
%T A380561 1,1,2,2,2,2,2,2,2,2,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,2,2,
%U A380561 1,2,2,2,2,2,2,2,2,2
%N A380561 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, 1, 2, 1, 1, 2, 1, 1, 2, ...) = (A100063(n)) for n >= 1. See Comments.
%C A380561 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 A380561 See A380560 for a guide to related sequences.
%e A380561 Corner:
%e A380561     1  2  1  2  1  1  2  1  2  1  1  2  1  2  1  2  1  1  2
%e A380561     1  2  2  1  2  2  1  2  1  1  2  1  1  2  1  2  2  1  2
%e A380561     2  1  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1  2
%e A380561     1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2  1
%e A380561     1  2  1  1  2  1  1  2  2  1  2  1  1  2  1  2  2  1  1
%e A380561     2  1  1  2  1  2  2  1  2  1  1  2  2  1  2  2  1  2  1
%e A380561     1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1
%e A380561     1  2  1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1
%e A380561     2  1  1  2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2
%t A380561 invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
%t A380561     Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
%t A380561 row1 = Flatten[Table[Nest[Flatten[# /. {1 -> {1, 2}, 2 -> {1, 1, 2}}] &, {1},  n], {n, 3}]];(* A006337 *)
%t A380561 rows = {row1};
%t A380561 col = PadRight[{}, 30, {1, 1, 2}];
%t A380561 Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
%t A380561 rows // ColumnForm  (* array *)
%t A380561 v[n_, k_] := rows[[n]][[k]];
%t A380561 Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*sequence*)
%t A380561 (* _Peter J. C. Moses_, Nov 20 2024 *)
%Y A380561 Cf. A000002, A100063 (column 1), A006337, A380560.
%K A380561 nonn,tabl
%O A380561 1,2
%A A380561 _Clark Kimberling_, Jan 27 2025