This content originally appeared on HackerNoon and was authored by Hockey Stick
:::info Author:
(1) Sima Mehri, Farzanegan High School.
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Table of Links
Abstract and 1 Introduction and Description of Results
2. Proof of Results and References
Abstract
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)
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\ 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)
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\ 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.
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\ Equivalently, the trinomial coefficients are defined by
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\ We have proven the following theorem in this triangle:
\ Theorem 1.3. (The Hockey Stick Theorem in The Trinomial Triangle)
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\ For Example see Figure 4.
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:::info This paper is available on arxiv under CC BY 4.0 DEED license.
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This content originally appeared on HackerNoon and was authored by Hockey Stick
Hockey Stick | Sciencx (2024-06-26T17:00:18+00:00) Exploring Hockey Stick Theorems: Abstract, Introduction and Description of Results. Retrieved from https://www.scien.cx/2024/06/26/exploring-hockey-stick-theorems-abstract-introduction-and-description-of-results/
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