Symbol merger

The total amount of symbols required to level from $k \in \mathbb{N}$ to $k + 1$ is $k^2 + 11$. So the total amount of symbols required to level from 1 to $n \in \mathbb{N}$ is $p(n) := \sum_{k=1}^n (k^2 +11).$

The degree four polynomial $p$ has a simpler and more computationally efficient expression. Its coefficients are given by
$ \begin{pmatrix} 1^3 & 1^2 & 1^1 & 1^0 \\ 2^3 & 2^2 & 2^1 & 2^0 \\ 3^3 & 3^2 & 3^1 & 3^0 \\ 4^3 & 4^2 & 4^1 & 4^0 \\ \end{pmatrix}^{-1} \begin{pmatrix} p(1) \\ p(2) \\ p(3) \\ p(4) \\ \end{pmatrix}. $