Calculate the number of times the pattern appears in a string

In this question, you are essentially being asked to create an algorithm that will count the number of times a certain pattern is found within a given string of characters. This is common in many algorithms and computer science contexts. There are few key points you need to keep in mind:

1. You need to understand what pattern is.
2. Think about how to iterate through the entire string efficiently.
3. Determine if the current subset of characters matches the given pattern.
4. Keep a count of how many times this match occurs.

Your algorithm should be able to handle cases where the pattern is larger than the string, or the pattern does not appear in the string at all.

For example, let's take the string "abcabcabc" and the pattern "abc". To find how many times "abc" occurs within the string, we would start by iterating through the input string, and for each subset of characters of the length of the pattern, we check if it matches the pattern. In this case, the pattern "abc"'s length is 3, so we would check every 3 characters in the string. Starting from index 0 to 2, we find "abc" which matches the pattern. Then we move on to the next 3 characters, 3 to 5, where we again find "abc". Finally, checking characters 6 to 8, we find a third "abc". Thus, the pattern "abc" appears 3 times in the string.

Let's consider another example with string being "abababab" and the pattern being "aba". Following the above algorithm, we would check every three characters in the string. Here, starting from index 0 to 2, we find "aba" which matches the pattern. Next, checking characters 2 to 4, we again find "aba". Similarly, on checking characters 4 to 6, we find the third "aba". Lastly, characters 6 to 8 give us the fourth "aba". So, the pattern "aba" appears 4 times within the string. Do note that overlapping patterns were considered separately in this algorithm, which is why we moved to the next character after finding a match, instead of skipping over the length of the pattern.