TI-Basic, 88 bytes

Input N
{1→A
While N≠sum(ʟA²
1+ʟA(1→ʟA(1
For(I,1,dim(ʟA
If N≤ʟA(I
Then
1→ʟA(I
If I<dim(ʟA
1+ʟA(I+1
Ans→ʟA(I+1
End
End
End
ʟA²

works with any number of panels (more than 4 if it was possible). It's basically an addition with a carry (up to N and starting at 1) because recursion is impossible a real pain in TI-Basic (all variables are global)

It is quite slow, it could be optimized for speed by setting the limit to √N instead of N and a condition such as ʟA(I+1)≤ʟA(I)