If you take a square of side

and a set of smaller squares of side

,

in

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

of them will fit.

If we place the small squares so that their bases are on the diagonal, then we are asking for given

to find the largest

such that

Then we may consider the sequence

.

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