Does this function have a complement on the positive integers in the sense that the range of the complement should be the complement of the range of the function?
$\frac{1}{8} \left(1-(-1)^n+2 n (2+n)\right)$
Here is the inverse/complement which I got from Generic Human:
$\left\lfloor \sqrt{4 n-1}\right\rfloor +n+1$
Edit per @robjohn:
Domain is $\mathbb{N}$. I'm trying to use the Lambek-Moser theorem to generate complimentary sets.
The formula generates the pronic numbers interleaved with the squares. $\{1,2,4,6,9,12,16,20,25,30\}$
I want to generate the complementary set of everything else.
$\{3,5,7,8,10,11,13,14,15,17\}$
The Lambek-Moser theorem uses the inverse to generate the complement.
My question is: am I interpreting this scenario properly?