Title: Exploring the Properties of the Honeycomb Sequence in Pascal's Triangle

Body:

I am studying a specific number sequence I am calling the the "Honeycomb Sequence," derived from the 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 in this area.

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 $N(n)$:

[ 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} ]

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:

[ N(n) = \left( \sum_{i=T_n + 1}^{T_{n+1}} i \right)^2 ]

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

1. Has there been any similar research or findings related to sequences formed from Pascal's Triangle in this manner?
2. What could be the mathematical explanation for the consistent digital root phenomenon observed in the Honeycomb Sequence?
3. Are there any known applications or implications of this sequence in other areas of mathematics or combinatorics?

I am eager to learn more about the properties of this sequence and any related mathematical concepts or theories. Any insights or references to relevant work would be greatly appreciated.