I am an amateur, and for fun, I was studying a specific number sequence I called the "Honeycomb Sequence," derived from hexagonal patterns in Pascal's Triangle. The sequence involves calculating products of binomial coefficients arranged in a hexagonal shape. I have observed intriguing properties in this sequence and seek insights or related work.
Hexagon Products and Honeycomb Sequence Formation
The formation of a hexagon in Pascal's Triangle is central to defining the Honeycomb Sequence. For each term $n$ greater than 1, the hexagon is constructed using binomial coefficients located at specific positions relative to $n$. The positions are:
- $\binom{n}{k+1}$
- $\binom{n+1}{k}$
- $\binom{n-1}{k-1}$
- $\binom{n-1}{k}$
- $\binom{n+1}{k+1}$
- $\binom{n}{k-1}$
These coefficients are multiplied to give the product $P(n)$, defining the sequence term $N(n)$. The sequence is constructed by calculating $P(n)$ for each term starting from $n = 2$, with $N(1)$ defined as 0.
Mathematical Proof of Sequence Properties
For specific terms, such as $N(3)$, we calculate the product of the binomial coefficients forming the hexagon. This calculation involves determining the values from Pascal's Triangle and multiplying them together. The product simplifies to a general expression for $N(n)$:
\begin{equation} P(n) = \binom{n}{k+1} \times \binom{n+1}{k} \times \binom{n-1}{k-1} \times \binom{n-1}{k} \times \binom{n+1}{k+1} \times \binom{n}{k-1}, \end{equation} with ( k = n-1 ). For $n = 3$, the product calculation yields $144$, a perfect square.
The Triangle of Consecutive Sums
An important characteristic of the Honeycomb Sequence is its alignment with a triangular pattern of sums, correlating each term $N(n)$ with the sum of a series of consecutive numbers. The formula for $N(n)$ using triangular numbers is: \begin{equation} N(n) = \left( \sum_{i=T_n + 1}^{T_{n+1}} i \right)^2 \end{equation}
This formula expresses $N(n)$ as the square of the sum of consecutive numbers starting from the number immediately after $T_n$ and ending at $T_{n+1}$.
Digital Root Phenomenon in the Honeycomb Sequence
An interesting observation is that the digital roots of the terms in the Honeycomb Sequence consistently add up to 9. This pattern holds for all values of $N(n)$ analyzed, with each term's digital root calculated by iteratively summing the digits until a single digit remains.
Questions and Discussions
- Has there been any similar research or findings related to sequences formed from Pascal's Triangle in this manner? ()
- Should I publish this in arXiv? Is it up to the standard of a mathematical journal?
- Suggestions for enhancing the mathematical depth or presentation of my findings.
- Advice on suitable academic journals or platforms where this work might be considered for publication.
I am also open to working together or guidance on further development of this work. Any insights or references to similar works would be greatly appreciated.
