Turing machine accepting words of "1" arrange-able into square

I'm trying to figure out following assignment and I could use your help, because I'm stuck. I'm not aiming for direct answer (but I don't mind if someone is capable of designing the whole machine). I'm more interested in approving/disproving my thoughts and pointing in the right directions.

Task:

Construct a Turing machine accepting words in format 1^k, where k=n^2 n is integer.Basically it accepts those words, where you can arrange ones into square.For example first three acceptable words will be:

1Δ; 1111Δ; 111111111Δ;

Which are arrange-able into squares like this:

11111111111111

My thoughts:

If I get "1" only once, it should be accepted.

Tape: 1Δ; k=1, n=1 and k=n^2

So I m assuming after one "1" I'm in the "STOP" state (lets call it state A) if there is no other char on the tape.

The next acceptable word is the next square, which we I can write in the formula like this:

(n+1)^2  That equals: n^2 + 2*n + 1

Difference between two consecutive squares is (2*n + 1)

So If my assumption is correct, If I'm in state A I need to somehow check if I get (2*n + 1) times "1" on tape and if that takes me back to state A, it will be correctly accepted. Anything else should be declined.

The problem I'm facing is, that I don't know, how to count (2*n + 1) times "1" on tape.

Also I should find solution using only one tape. The hint is that it should end in the form111#111111111Δ, where the left side sum of ones equals "k" and right side equals "n". Sadly, this hint didn't help me at all. I'd say it made things worse for me.