Note: Analogous to the golden ratio, the bronze ratio is the positive solution to the equation x2=3x+1, which equals 23+32+4​​ and has the continued fraction expansion 3+3+3+…1​1​. The approximation evaluated in this problem is accurate to three decimal places (3.303). Assuming the continued fraction expansion has a limiting value x0​, it can be seen that x0​=3+x0​1​, which is equivalent to x02​=3x0​+1.
The bronze ratio, 23+32+4​​, is analogous to the golden ratio, 21+12+4​​. The golden ratio is associated with
1+1+1+⋯1​1​
Similarly, the silver ratio is 22+22+4​​, associated with
2+2+2+⋯1​1​.
These metallic ratios are also related to generalizations of the Fibonacci sequence, arise as lengths of diagonals in regular polygons, and have many other geometric and algebraic interpretations as well.