If you take a square of side $\displaystyle $$1$ and a set of smaller squares of side $\displaystyle $$1/n$, $\displaystyle $$n$ in $\displaystyle $$\mathbb{N}$ then how many of these smaller squares can be fitted on the diagonal.

This is still ambiguous, and what we mean by "fitted on the diagonal" is not clear.

If they are placed so the small square diagonals lie on the unit squares diagonal obviously $\displaystyle $$n$ of them will fit.

If we place the small squares so that their bases are on the diagonal, then we are asking for given $\displaystyle $$n$ to find the largest $\displaystyle $$k_n$ such that $\displaystyle $$(k_n/n)^2\le 2$

Then we may consider the sequence $\displaystyle k_n,\ n=1...$.

The following calculates the first few terms, which seems to agree with what you post

Code:

>function seq(n)
$ s=zeros(n,2);
$ for idx=1 to n
$ s(idx,2)=floor(sqrt(2)*idx);
$ s(idx,1)=idx;
$ end
$ return s
$endfunction
>
>seq(10)
1 1
2 2
3 4
4 5
5 7
6 8
7 9
8 11
9 12
10 14
>

Note this is still largely guess work since your clarification still makes little sense.

CB