How to connect the math of recurrence relations to daily programming concepts
In computer science, recurrence relations are often useful for analyzing the running time of algorithms.  For instance, we might define a function $T(\cdot)$, so that $T(n)$ is the running time when running the algorithm on an input of size $n$.  Often, for many kinds of algorithms, we can write a recurrence relation that expresses $T(n)$ in terms of smaller values, such as

$$T(n) = 2T(n/2) + O(n).$$

This would be saying that the running time to solve a problem of size $n$, is twice the time to solve a problem of half that size (perhaps because we recursively solve the first half and the second half using the same algorithm), plus $O(n)$ more (perhaps to combine/merge their two solutions).

So, the numbers represent running times -- the number of operations the algorithm takes to solve the problem.  You might like to take a look at Reference answers to frequently asked questions and How to come up with the runtime of algorithms?.  Also take a look at algorithms textbooks, especially their chapters on running time analysis and divide-and-conquer algorithms.

The initial condition of a recurrence (e.g., the value of $T(1)$) depicts the running time of the algorithm on a small instance.  Often, for a recursive algorithm, this is the running time of the base case.

In computer science, the sequence we are considering is

$$T(1),T(2),T(3),T(4),T(5),\dots,$$

i.e., the sequence of running times as the input size increases.  If you were to write down the sequence of running times for some algorithm, you might spot a pattern, where each term can be computed by a simple formula of a few previous terms (namely, the pattern indicated by the recurrence relation).  That said, honestly, thinking about this as a sequence might not be the most helpful or intuitive viewpoint (at least, I don't find it to be so); I find it more intuitive to think of $T(\cdot)$ as a function rather than a sequence.  The same mathematics still applies.

Another way to get some intuition is to work through a few examples where recurrences are used to analyze the running time of some simple algorithms.  I think that will also help you get a better intuition of what the mathematical expressions mean and how they relate to the algorithm.