O(n) Palindromic Substring: Why is O(n^2) needed here?

Question

I have been doing dynamic programming problems, and I came across a palindromic substring question. I answered it quickly, and found an $O(n)$ dynamic programming solution, but when I check the actual solution, I find that I should be getting an $O(n^2)$ result.

Can someone explain where I am going wrong/how this is not correct?

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The problem reads: given string $x_1,x_2,\ldots,x_n$, find the longest palindromic substring.

I write:

$$ T(i) = \text{max length palindromic substring ending at position }i\text{ in string }X $$

Then I write down the following subproblem (where $j$ is intended to be the letter right before the beginning of the palindrome ending at position $i$):

$$ \begin{align} \text{IF }x_i = x_j, \text{ where } &j = i - T(i-1)-1:\\ T(i) &= T(i - 1) + 2;\\ \text{ELSE IF} x_i = x_{i-1} \\ T(i) &= 2;\\ \text{ELSE } T(i) &=1; \end{align} $$

As far as I can tell, this is correct. In order to recover the palindrome, I record it in linked lists or whatever, and calculate the argmax of $T$.

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Where am I going wrong?