Number Theory

    

Unexpected Connection Between Triangular Numbers and the Golden Ratio

Authors: Waldemar Puszkarz

We find out that when a sum of five consecutive triangular numbers, $S_5(n)= T(n)+...+T(n+4)$, is also a triangular number $T(k)$, the ratios of consecutive terms of $a(i)$ that represent values of $n$ for which this happens, tend to $phi^2$ or $phi^4$ as $i$ tends to infinity, where $phi$ is the Golden Ratio. At the same time, the ratios of consecutive terms $S_5(a(i))$ tend to $phi^4$ or $phi^8$. We also note that such ratios that are the powers of $phi$ can appear in the sequences of triangular numbers that are also higher polygonal numbers, one case of which are the heptagonal triangular numbers.

Comments: 5 Pages. Originally posted on ResearchGate in March 2023.

Download: PDF

Submission history

[v1] 2023-08-10 23:45:22

Unique-IP document downloads: 186 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus