Infinite Well: Parity

The wave functions have the symmetry property

$$\phi_n(x+L/2)=(-1)^{n-1}\phi_n(L/2-x),$$ (254)

that is they are symmetric or antisymmetric with respect to the symmetry point $x=L/2$ of the infinite well potential $V(x)$, in alternating order with the quantum number $n$. The lowest eigenstate $\phi_1(x)$ is symmetric and has no node, the next eigenstate $\phi_2(x)$ is anti-symmetric and has one node, $\phi_3(x)$ is again symmetric and has two zeros, and so on.

We recognize that conceptually, everything is really completely analogous (the counting from 0 or $1$ is a matter of convention).