Zeckendorf explored the decomposition of integers into sums of nonconsecutive Fibonacci numbers. Thinking of each Fibonacci number in the sequence as being inside a “bin”, Zeckendorf’s rule for “legal” decompositions can be reworded as sums of numbers in bins separated by one or more bins. By generalizing these ideas, we create new sequences. Specifically, the bins can contain different quantities of sequence terms and “legal” decompositions are redefined to require more bins between summands. These sequences can be formed by a single recurrence relation, similar to the Fibonacci sequence. It turns out that given different initial conditions, very similar sequences are produced. We discuss where and why the same terms appear. Additionally, we examine the ratios of consecutive numbers as well as quotients of summands. Our results show that ratios made up of terms coming from the same relative positions are approximately equal.