An Application of a Theorem I Just Learned

Question : Recall that a polygonal number is a number that counts dots arranged in the shape of a regular polygon. Prove that the $n$-th non $s$-gonal number is given by

$$\left\lfloor \frac 12 + \sqrt{\frac 14 + \frac{2n}{s-2}}\right\rfloor + n$$

where $\lfloor x\rfloor$ represents the least integer less than or equal to $x$.

Answer : It is well known that

$$(s-2)\frac{n(n-1)}{2}+n$$

is the $n$-th $s$-gonal number.

Now, we will use the Lambek-Moser Theorem to get

$$f(n)=(s-2)\frac{n(n-1)}{2}$$

and hence

$$f^\ast (n) = \left\lfloor \frac 12 + \sqrt{\frac 14 + \frac{2n}{s-2}}\right\rfloor$$

and hence

$$F^\ast (n) = \left\lfloor \frac 12 + \sqrt{\frac 14 + \frac{2n}{s-2}}\right\rfloor + n$$

hence completing the proof.

Remark : A careful induction on $n$ (keeping $s$ fixed) would probably have also done the job.

Acknowledgement : I learnt about the Lambek-Moser Theorem from the book The Irrationals: a Story of the Numbers You Can't Count On by Julian Havil (section Theoretical Matters of chapter Does Irrationality Matter?) which I also reviewed for zbMATH.