Exploring Hockey Stick Theorems: Abstract, Introduction and Description of Results

26 Jun 2024


(1) Sima Mehri, Farzanegan High School.

Abstract and 1 Introduction and Description of Results

2. Proof of Results and References


There are some theorems in the Pascal’s triangle which their figures resemble to shoot a ball by hockey stick, so they are called hockey stick theorems. P. Hilton and J. Pedersen, in the article ”Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry”,[2], have stated the little and big hockey stick and puck theorems in the Pascal’s triangle. The big hockey stick theorem is a special case of a general theorem which our goal is to introduce it. We state a hockey stick theorem in the trinomial triangle too.

1 Introduction and Description of Results

The big hockey stick and puck theorem, stated in [2] is:

Theorem 1.1. [2] (The Big Hockey Stick and Puck Theorem)

In [2], this theorem is also demonstrated by Figure 1. We have found the general form of above theorem in Pascal triangle as below.

Theorem 1.2. (The Hockey Stick Theorem in Pascal Triangle)

An example of this theorem is illustrated in Figure 2.

Now we wish to state the hockey stick theorem in trinomial triangle. First using [3] and [4], we explain what is the trinomial triangle.

Figure 1: The Big Hockey Stick and Puck Theorem

Figure 2: Example of Hocky-Stick:1+3+10+35=56-7

Figure 3: Trinomial Coefficients

Equivalently, the trinomial coefficients are defined by

We have proven the following theorem in this triangle:

Theorem 1.3. (The Hockey Stick Theorem in The Trinomial Triangle)

For Example see Figure 4.

This paper is available on arxiv under CC BY 4.0 DEED license.