Ok so what I found was a square matrix of order $n×n$ where $n$ follows $2m+1$ and $m$ is a natural number
the pattern these matrices follow is as follows:
for a $3×3$ matrix: $$ A = \left( \begin{array}{ccc} b & a & c \\ a & a+b+c & a \\ c & a & b \end{array} \right)$$ and the cool thing is that $$ A^x = \left( \begin{array}{ccc} q & p & r \\ p & p+q+r & p \\ r & p & q \end{array} \right)$$ where $ x $ is any natural number
Now for $5×5$ it goes like $$ A = \left( \begin{array}{ccc} b & b & a & c & c\\ b & b & a & c & c\\ a & a & 2b+a+2c & a & a\\ c & c & a & b & b\\ c & c & a & b & b\end{array} \right)$$ and again $$ A^x = \left( \begin{array}{ccc} q & q & p & r & r\\ q & q & p & r & r\\ p & p & 2q+p+2r & p & p\\ r & r & p & q & q\\ r & r & p & q & q\end{array} \right)$$
Once more for a $7×7$ matrix we have $$ A = \left( \begin{array}{ccc} b & b & b & a & c & c & c\\ b & b & b & a & c & c & c\\ b & b & b & a & c & c & c\\ a & a & a & 3b+a+3c & a & a & a\\ c & c & c & a & b & b & b\\ c & c & c & a & b & b & b\\ c & c & c & a & b & b & b\\ \end{array} \right)$$ and then $$ A^x = \left( \begin{array}{ccc} q & q & q & p & r & r & r\\ q & q & q & p & r & r & r\\ q & q & q & p & r & r & r\\ p & p & p & 3q+p+3r & p & p & p\\ r & r & r & p & q & q & q\\ r & r & r & p & q & q & q\\ r & r & r & p & q & q & q\\ \end{array} \right)$$
and so on and so forth, I didn't find whether this has already been observed and neither a name for this so I am calling this checkboard matrix, because it looks like that.
I just have one question, has this been found already, if yes please give details.
Addendum: Thanks to Robert Israel for pointing out the error, I have made some changes, please check into it. The pattern for the middle element is $ m (b+c)+a $