I have these two sequences $ a,b $
$$ \begin{array}{c|lcr} n & \text{a} & \text{b} \\ \hline 1 & 1 & 0 \\ 2 & 2 & 1 \\ 3 & 2 & 0 \\ 4 & 3 & 2 \\ 5 & 3 & 1 \\ 6 & 3 & 0\\ 7 & 4 & 3 \\ 8 & 4 & 2 \\ 9 & 4 & 1 \\ 10 & 4 & 0 \\ 11 & 5 & 4 \\ \vdots & \vdots & \vdots \end{array} $$
what should I do to write a function using $a$ and $b$ to get $n$? Can you please show me some direction? Thanks.
$$n = {a+1\choose2} - b$$ How do we get that?
Given $a$, we conclude $$ 1+2+\cdots+a \geq n > 1+2+\cdots+(a-1) $$ where $$ 1+2+\cdots+a = {a+1\choose2} $$ by a "well known" formula. Now, looking at $b$, we see that all we have to do is subtract $b$ from ${a+1\choose2}.$
There you are.