A380561 -id:A380561 - cp's OEIS frontend

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.

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, 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, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1

Offset: 1

Clark Kimberling, Jan 27 2025

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.
Row 1: A006337; limiting row: A000002 (Kolakoski sequence).
Guide to related sequences (arrays):
A378282, limiting sequences of periodic inverse runlength arrays
A380560, (row 1)=A006337, periodic (column 1)=(1,...)
A380561, (row 1)=A006337, periodic (column 1)=(1,1,2,...)
A380562, (row 1)=A006337, periodic (column 1)=(1,2,2,...)
A380563, (row 1)=1+A010060, periodic (column 1)=(1,...)
A378303, (ro1 1)=A006337, periodic (column 1)=(2,2,1,...)
A378396, self-inverse runlength array, u = v = 1+A010060
A378397, self-inverse runlength array, u = v = A003842
A378398, self-inverse runlength array, u = v = A014675
A378399, self-inverse runlength array, u = v = A006337

			Corner:
  1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2 1 1 2 1
  1 2 2 1 2 2 1 2 1 1 2 1 1 2 1 2 2 1 2 2
  1 2 2 1 1 2 1 1 2 2 1 2 2 1 2 1 1 2 1 2
  1 2 2 1 1 2 1 2 2 1 2 1 1 2 2 1 2 2 1 1
  1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 1 2
  1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1
  1 2 2 1 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1

Cf. A000002, A000012 (column 1), A006337.

invR[seq_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {1, 2}, {2, -1, 2}], 2]]];
s = Differences[Table[Floor[n*Sqrt[2]], {n, 1, 21}]]; (* A006337 *)
t = NestList[invR, s, 12];
u[n_] := Take[t[[n]], 20];
Table[u[n], {n, 1, 12}]  (* array *)
v[n_, k_] := t[[n]][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Nov 13 2024 *)

cp's OEIS Frontend

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.

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