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by Hockey StickJune 26th, 2024

**Author:**

(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.

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.

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.

L O A D I N G

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