Over a million developers have joined DZone.
{{announcement.body}}
{{announcement.title}}

DZone's Guide to

Golden Strings and the Rabbit Constant

· Big Data Zone ·
Free Resource

Comment (0)

Save
{{ articles[0].views | formatCount}} Views

The open source HPCC Systems platform is a proven, easy to use solution for managing data at scale. Visit our Easy Guide to learn more about this completely free platform, test drive some code in the online Playground, and get started today.

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

• “1″
• “10″
• “101″
• “10110″
• “10110101″

The length of sn is Fn+1, the n+1st Fibonacci number. Also, sn contains Fn-1 1′s and Fn-2 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 sympy.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.

Managing data at scale doesn’t have to be hard. Find out how the completely free, open source HPCC Systems platform makes it easier to update, easier to program, easier to integrate data, and easier to manage clusters. Download and get started today.

Topics:

Comment (0)

Save
{{ articles[0].views | formatCount}} Views

Published at DZone with permission of

Opinions expressed by DZone contributors are their own.