Well defined sequence explanation Can anyone explain what is meant by this sentence on the following wikipedia article. I don't see what they are trying to explain here. It is the last paragraph in the Indexing section.

In the second and third bullets, there is a well-defined sequence $(a_{k})_{k=1}^{\infty}$, but it is not the same as the sequence denoted by the expression.

EDIT:
By the answers I got it seems that 2nd and 3rd bullets are not enough to define a sequence of squares of odd numbers. How is it then enough in the 4th bullet by just writing this $(a_{k})_{k=1}^{\infty}$?
 A: In the second and third bullets, $a_k=k^2=(1,4,9,\dots)$, but it is not the same as the sequence they are trying to define, i.e. the sequence of squares of odd numbers.
The author's point is you have to consider both parts of the expression for the second and third bullets: $(a_1, a_3, a_5, \dots)$ and $a_k=k^2$, or equivalently, $(a_{2k-1})_{k=1}^\infty$ and $a_k=k^2$, to accomplish defining the sequence of interest.
Note: the difference between the second and third bullets versus the fourth bullet is that $a_k$ in the fourth bullet is in fact equal to the sequence of squares of odd numbers, but in the second and third bullet, $a_k$ is a different sequence, so the author warns of not taking $a_k$ as the sequence of interest, despite however natural it might be to do so.
A: It means that the target sequence is defined through another sequence. 
The second bullet defines the sequence $(x_1,x_2,x_3,\dots)=(1,9,25,49,\dots)$ odd squares by first setting $a_n:=n^2$, which is $(1,4,9,16,\dots)$, then considering a subsequence of it, keeping every second element only:
$$(a_1,a_3,a_5,a_7,\dots)$$
The note says that we don't directly have $x_n=a_n$. 
