Function of the following sequence

Question

I feel like this should be obvious for me, but I am not understanding this one. Given: 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,..., I need to find a function for this. Thank you.

Answer 1

The On-Line Encyclopedia of Integer Sequences gives several formulas, including $$ a_n=\left\lfloor\frac{1}{2}+\sqrt{2n}\right\rfloor $$

The main point is that the sequence changes values at positions $1,2,4,7,11,\dots$, which is the sequence of triangular numbers $0,1,3,6,10,\dots$ shifted $1$ and these triangular numbers are given by $\frac{k(k-1)}{2}$. Hence the sequence of positions where the original sequence changes values is given by $j_k=\frac{k(k-1)}{2}+1$, $k=1,2,\dots$.

So the original sequence is given by $a_n=k$ if $j_k \le n < j_{k+1}$.

Answer 2

f(x) = if (x mod f(x - 1) == 0) 
         then 
           x mod f(x - 2) + 1 
        else 
           x mod f(x - 1)  

Only I don't know how to put if in to a math function.