16 is the smallest 2digit square whose digits are nondecreasing.
144 is the smallest 3digit square whose digits are nondecreasing.
1156 is the smallest 4digit square whose digits are nondecreasing.
GO ON!
You might think that beyond a certain length, the sequence must end but it does not.
Although not always the smallest, note the sequences:
34^2=1156
334^2=111556
3334^2=11115556
which covers all the even lengths except 2
17^2=289
167^2=27889
1667^2=2778889
which covers all the odd lengths
The above are sometimes, but not always, the smallest.
The actual sequence begins
16 = 4^2
144 = 12^2
1156 = 34^2
11236 = 106^2
111556 = 334^2
2666689 = 1633^2
11115556 = 3334^2
277788889 = 16667^2
1111155556 = 33334^2
11122233444 = 105462^2
the next few terms use the sequences I gave at the top.
see https://oeis.org/A028820/b028820.txt
which lists all squares with increasing terms but is easy to scan for the smallest of each length.

Posted by Jer
on 20150512 09:36:57 