$$a_n=a_0+2n$$ $$S_p=a_0+a_1+...+a_{p-1}$$ In $S_p$ is $a_0+a_{p-1}=a_n+a_{(p-1)-n}=C$. Thus every pair $a_n, a_{(p-1)-n}$ can be expressed as a constant value $C$. In $S_p$ there exist $\frac{1}{2}p$ pairs $a_n, a_{(p-1)-n}$.
Hence $S_p=\frac{1}{2}pC, p\ge2$ and $S_p=a_0, p=1$ and $S_p=undefined, p=0$.
However I find it not intuitive that for $p=3$ there exist $1,5$ pairs. I tried to express the problem geometrically instead of using the word "pair." Which rule of mathematics can I use to say that despite my lack of intuition according to mathematics the above is correct?