S = start symbol
A = first left bookend
B = second left bookend
C = third left bookend
D = first right bookend
E = second right bookend
X = first token
R = runner
Y = second token (moves left)
Z = third token (moves right)

S -> AD       generate first bookends
A -> AX | B   generate n X's for some n, then turn into second bookend
B -> CR       generate runner
RX -> YXR     generate one Y for each X, n Y's altogether
RD -> D       absorb runner
XY -> YX      move Y leftward past X
ZY -> YZ      move Y leftward past Z
CY -> CZ      each Y changes to Z, n Z's altogether
ZX -> 1XZ     generate 1 each time a Z passes an X, n^2 1's altogether
Z1 -> 1Z      move Z rightward past 1
ZD -> D       absorb Z after it's done its job
D -> E        turn into second bookend
X1 -> 1X      move X rightward past 1
XE -> E       absorb X
C -> empty    erase left bookend
E -> empty    erase right bookend

Here's how to generate strings for n = 0, n = 1, and n = 2:

S => AD => BD => CRD => CD => CE => E => empty 

S => AD => AXD => BXD => CRXD => CYXRD => CYXD => CZXD => C1XZD => C1XD
  => C1XE => C1E => 1E => 1

S => AD => AXD => AXXD => BXXD => CRXXD => CYXRXD => CYXYXRD => CYXYXD
  => CZXYXD => CZYXXD => CYZXXD => CZZXXD => CZ1XZXD => CZ1X1XZD => CZ1X1XD
  => C1ZX1XD => C11XZ1XD => C11X1ZXD => C11X11XZD => C11X11XD => C11X11XE
  => C11X11E => C111X1E => C1111XE => C1111E => 1111E => 1111