Counting Fibonacci Squares

Consider the above sequence. In step \(n\), we add \( F_n \) (Fibonacci number) unit squares in a column to the right. We want to count the number of squares (of all sizes) in each figure.

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

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

If \(n = 3\), then 2 small squares are added, so we count 4 1-by-1 squares.

If \(n = 4\), then 3 small squares are added, so we count 7 1-by-1 squares and a 2-by-2 square, which altogether makes 8 squares in all.

For \(n = 5\), is the number of all possible different-sized squares 16?


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