From _Gary W. Adamson_, May 18 2021: 
As an irregular triangle (n, A001511(k)) = 1; otherwise 0:
  1
  0 1
  1
  0 0 1
  1
  0 1
  1
  0 0 0 1
  ...
Consider the above as an infinite triangular matrix M, with any sequence as a single column vector V, where the terms in V represent Tower of Hanoi (TOH) disks (1, 2, 3, ...) relabeled with the target sequence.  Then M * V returns the corresponding TOH sequence.  Examples:
Given V = (1, 2, 3,  4, ...), then M*V = (1, 2, 1, 3, 1, 2, 1,  4, ... =  A001511.
......V = (1, 3, 6, 10, ...), then M*V = (1, 3, 1, 6, 1, 3, 1, 10, ...)
......V = (1, 2, 4,  8, ...), then M*V = (1, 2, 1, 4, 1, 2, 1,  8, ...) = A066519
......V = (1, 0, 1,  0, ...), then M*V = (1, 0, 1, 1, 1, 0, 1,  0, ...) = A035263
For the last result, A035263 can be defined as TOH(1, 0 1, 0, ...), or the Tower of Hanoi sequence using disks labeled (1, 0, 1, 0, ...). The operation is subject to the usual optimal moves on TOH pegs (0, 1, 2); where disks labeled "1" move CW on pegs (0, 1, 2, 0, 1, 2, ...), and disks labeled 0 move CCW in the orbit (2, 1, 0, 2, 1, 0, 2, 1, 0, ...).