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A380560 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 for n >=1, See Comments.

%I A380560 #7 Feb 06 2025 12:37:32
%S A380560 1,2,1,1,2,1,2,2,2,1,1,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,2,1,1,1,2,1,1,2,
%T A380560 2,1,2,2,1,2,1,1,2,2,1,1,1,1,1,2,1,1,2,2,1,1,1,2,2,1,2,1,1,2,2,1,2,2,
%U A380560 2,2,2,1,2,1,1,2,2,1
%N A380560 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 for n >=1,  See Comments.
%C A380560 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 consists solely of 1s. No two rows are identical.
%C A380560 Row 1: A006337; limiting row: A000002 (Kolakoski sequence).
%C A380560 Guide to related sequences (arrays):
%C A380560 A378282, limiting sequences of periodic inverse runlength arrays
%C A380560 A380560, (row 1)=A006337, periodic (column 1)=(1,...)
%C A380560 A380561, (row 1)=A006337, periodic (column 1)=(1,1,2,...)
%C A380560 A380562, (row 1)=A006337, periodic (column 1)=(1,2,2,...)
%C A380560 A380563, (row 1)=1+A010060, periodic (column 1)=(1,...)
%C A380560 A378303, (ro1 1)=A006337, periodic (column 1)=(2,2,1,...)
%C A380560 A378396, self-inverse runlength array, u = v = 1+A010060
%C A380560 A378397, self-inverse runlength array, u = v = A003842
%C A380560 A378398, self-inverse runlength array, u = v = A014675
%C A380560 A378399, self-inverse runlength array, u = v = A006337
%e A380560 Corner:
%e A380560   1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2 1 1 2 1
%e A380560   1 2 2 1 2 2 1 2 1 1 2 1 1 2 1 2 2 1 2 2
%e A380560   1 2 2 1 1 2 1 1 2 2 1 2 2 1 2 1 1 2 1 2
%e A380560   1 2 2 1 1 2 1 2 2 1 2 1 1 2 2 1 2 2 1 1
%e A380560   1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 1 2
%e A380560   1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1
%e A380560   1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1
%t A380560 invR[seq_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
%t A380560     Partition[Riffle[seq, {1, 2}, {2, -1, 2}], 2]]];
%t A380560 s = Differences[Table[Floor[n*Sqrt[2]], {n, 1, 21}]]; (* A006337 *)
%t A380560 t = NestList[invR, s, 12];
%t A380560 u[n_] := Take[t[[n]], 20];
%t A380560 Table[u[n], {n, 1, 12}]  (* array *)
%t A380560 v[n_, k_] := t[[n]][[k]];
%t A380560 Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
%t A380560 (* _Peter J. C. Moses_, Nov 13 2024 *)
%Y A380560 Cf. A000002, A000012 (column 1), A006337.
%K A380560 nonn,tabl
%O A380560 1,2
%A A380560 _Clark Kimberling_, Jan 27 2025