Counting Fibonacci Squares

Consider the diagram and the following pattern of counting all different-sized squares in increasing order:

If \(n = 1\), then we count 1 square.

If \(n = 2\), then 1 small square is added, so we count 2 squares in all.

If \(n = 3\), then 2 small squares are added, so we count 4 squares in all.

If \(n = 4\), then 3 small squares are added, so we count 8 squares in all, including a 2-by-2 square.

If we add the next Fibonacci number of small squares for \(n = 5\), is the number of all possible different-sized squares 16?

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