  In previous work, the second author and M\"uller determined the function
$c(a)$ giving the smallest dilate of the polydisc $P(1,1)$ into which the
ellipsoid $E(1,a)$ symplectically embeds. We determine the function of two
variables $c_b(a)$ giving the smallest dilate of the polydisc $P(1,b)$ into
which the ellipsoid $E(1,a)$ symplectically embeds for all integers $b
\geqslant 2$.
  It is known that for fixed $b$, if $a$ is sufficiently large then all
obstructions to the embedding problem vanish except for the volume obstruction.
We find that there is another kind of change of structure that appears as one
instead increases $b$: the number-theoretic "infinite Pell stairs" from the
$b=1$ case almost completely disappears (only two steps remain), but in an
appropriately rescaled limit, the function $c_b(a)$ converges as $b$ tends to
infinity to a completely regular infinite staircase with steps all of the same
height and width.
