Golden strings are analogous to Fibonacci numbers, except one uses concatenation rather than addition.
Start with s1 = “1” and s2 = “10”. Then define sn = sn-1 + sn-2 where “+” means string concatenation.
The first few golden strings are
The length of sn is Fn+1, the n+1st Fibonacci number. Also, sn contains Fn 1’s and Fn-1 0’s. (Source: The Glorious Golden Ratio).
If we interpret the sn as the fractional part of a binary number, the sequence converges to the rabbit constant R = 0.7098034428612913…
It turns out that R is related to the golden ratio φ by
where ⌊i φ⌋ is the largest integer no greater than iφ.
Here’s a little Python code to print out the first few golden strings and an approximation to the rabbit constant.
from mpmath import mp, fraction a = "1" b = "10" for i in range(10): b, a = b+a, b print(b) n = len(b) mp.dps = n denom = 2**n num = int(b, 2) rabbit = fraction(num, denom) print(rabbit)
Note that the code sets the number of decimal places,
mp.dps, to the length of the string
b. That’s because it takes up to n decimal places to exactly represent a rational number with denominator 2n.