For a long time, my friend, Advay Patil, and I have been investigating linear recurrences, and in this blog post, I will briefly detail extracts from our research. You can find a complete paper here.
The Fibonacci Sequence and Phi
Let us begin at a point of reference that many will have heard of. The Fibonacci sequence is defined by the recrursive formula:
\[F_n = F_{n-1} + F_{n-2}\]This recursive formula generates a sequence of integers (given the starting values of \(1, 1\)) that have a ratio that approaches the value \(\varphi\). \(\varphi\) can be described in many ways, and it is known that
\[\varphi^2 - \varphi - 1 = 0\]Which we can then use to state:
\[\varphi^2 - \varphi - 1 = 0\] \[\varphi^2 = \varphi + 1\] \[\varphi = \pm \sqrt{1 + \varphi}\] \[\varphi = \pm \sqrt{1 \pm \sqrt{1 \pm \varphi}}\] \[\varphi = \pm \sqrt{1 \pm \sqrt{1 \pm \sqrt{\cdots}}}\]This is just the first in a set of many interesting open forms which can be described, here is another:
\[\varphi^2 - \varphi - 1 = 0\] \[\varphi - 1 - \varphi^{-1} = 0\] \[\varphi^{-1} = \varphi - 1\] \[\varphi^{-1} = \varphi\left(1 - \varphi^{-1}\right)\] \[\frac{\varphi^{-1}}{1 - \varphi^{-1}} = \varphi\] \[\varphi = \varphi^{-1}(1 + \varphi^{-1} + \varphi^{-2} + \cdots)\] \[\varphi = (\varphi^{-1} + \varphi^{-2} + \varphi^{-3} + \cdots)\] \[\varphi = \sum_{n=1}^{\infty} \varphi^{-n}\]We are able to go from the fraction to the infinite sum using the sum of an infinite geometric series forumua. The ratio here being \(\varphi^{-1}\), given \(\varphi > 1\), \(\varphi^{-1} < 1\), allowing us to use that theorem.
The Plastic Number and the Supergolden Ratio
Here we shall head into the more obscure territory. Firstly, the Supergolden ratio (\(\psi\)). The Supergolden Ratio is the ratio found through the Sequence \(F_n = F_{n-1} + F_{n-3}\). It can be expressed through the polynomial \(\psi^3 - \psi^2 - 1 = 0\) and using similar logic to \(\varphi\), we are able to prove that it has this open form1:
\[\psi = \sum_{n=2}^{\infty} \psi^{-n}\]The introduction to the Supergolden ratio, while brief here, is important to the ideas of the Glass number.
The second value we shall take a look at is the Plastic number (\(\rho\)). The Plastic number is defined by \(\rho^3 - \rho - 1 = 0\), and is actually the ratio approximated by the sequence \(F_n = F_{n-1} + F_{n-5}\)2. From this we are able to quickly see that we can generate two open forms:
\[\rho^3 = \rho + 1\] \[\rho = \sqrt[3]{\rho + 1}\] \[\rho = \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{\cdots}}}\]And
\[\rho = \rho^3 - 1\] \[\rho = (\rho^3 - 1)^3 - 1\] \[\rho = ((((\cdots)^3 - 1)^3 - 1)^3 - 1)^3 - 1\]Note that the logic for the second open form may be applied to \(\varphi\) as well.
The Glass Number
Now we come to the crux of this blog post, the Glass number (\(\varrho\)). The Glass number is the reciprocal of the Supergolden Ratio, and as such had some interesting properties. But first, we need to derive the polynomials which describe it. Looking back on the open form of the Supergolden Ratio, we can create a polynomial to describe the Glass number.
\[\psi = \sum_{n=2}^{\infty} \psi^{-n}\] \[\frac{1}{\varrho} = \sum_{n=2}^{\infty} \frac{1}{\varrho^{-n}}\] \[1 = \varrho\left(\sum_{n=2}^{\infty} \varrho^{n}\right)\] \[1 = \varrho\left(\varrho^2 + \varrho^3 + \cdots\right)\] \[1 = \varrho^3\left(1 + \varrho + \varrho^2 + \cdots\right)\] \[1 = \varrho^3\left(\frac{1}{1 - \varrho}\right)\] \[1 = \frac{\varrho^3}{1 - \varrho}\] \[1 - \varrho = \varrho^3\] \[\varrho^3 + \varrho - 1 = 0\]Now, from here we are able to derive two open forms in a similar manner to those of the Plastic number, those being:
\[\varrho = \sqrt[3]{1 - \sqrt[3]{1 - \sqrt[3]{\cdots}}}\]And
\[\varrho = 1 - (1 -(1- (1 - (\cdots)^3)^3)^3)^3\]From this we can see that the Glass number is very similar to the Plastic number, however, Advay and I were unable to find anything more substantial about it, hopefully those reading this post may know more about how this is useful.