Centaur

A centaur is a chess piece that moves both as a knight (in an $L$-shape) and as a bishop (diagonally).

The possible moves of the centaur are illustrated in the diagram below (the position of the centaur is marked with a circle):

Consider an n \times n chessboard. The rows are numbered from $1$ to $n$ from top to bottom, and the columns are numbered from $1$ to $n$ from left to right. The square at the intersection of the $i$-th row and the $j$-th column is denoted as $(i,j)$.

The centaur is located on an n \times n board at square $(i,j)$, where 1 \leq i, j \leq n. Determine how many different squares the centaur can move to, according to its rules of movement.