HockeyStack

hockeystack

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(1) Sima Mehri, Farzanegan High School. Table of Links Abstract and 1 Introduction and Description of Results 2. Proof of Results and References 2. Proof of Results In the proof of both theorems, we use induction. using properties of Pascal triangle, we get The statement for k + 1 is also true, and the proof is completed. using properties of the trinomial coefficients, we get The statement for k + 1 is also true, and the proof is completed. The hockey stick theorem in the trinomial triangles has been proved. This theorem can be translated in Pascal pyramid as follows : Other similar theorems might be obtained for Pascal’s four dimensional and even n-dimensional pyramid. References 1] G. Andrews, Euler’s ’Exemplum Memorabile Inductionis Fallacis’ and Trinomial Coefficients J. Amer. Math. Soc. 3 (1990), 653-669. [2] P. Hilton and J. Pedersen, Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry Mathematics Magazine, Vol. 60, No. 5 (Dec., 1987), 305-316. [3] Eric W.Weisstein, Trinomial Coefficient From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html [4] Eric W.Weisstein, Trinomial Triangle From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) Sima Mehri, Farzanegan High School. Author: Author: (1) Sima Mehri, Farzanegan High School. Table of Links Abstract and 1 Introduction and Description of Results Abstract and 1 Introduction and Description of Results 2. Proof of Results and References 2. Proof of Results and References 2. Proof of Results In the proof of both theorems, we use induction. using properties of Pascal triangle, we get The statement for k + 1 is also true, and the proof is completed. using properties of the trinomial coefficients, we get The statement for k + 1 is also true, and the proof is completed. k The hockey stick theorem in the trinomial triangles has been proved. This theorem can be translated in Pascal pyramid as follows : Other similar theorems might be obtained for Pascal’s four dimensional and even n-dimensional pyramid. References 1] G. Andrews, Euler’s ’Exemplum Memorabile Inductionis Fallacis’ and Trinomial Coefficients J. Amer. Math. Soc. 3 (1990), 653-669. [2] P. Hilton and J. Pedersen, Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry Mathematics Magazine, Vol. 60, No. 5 (Dec., 1987), 305-316. [3] Eric W.Weisstein, Trinomial Coefficient From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html [4] Eric W.Weisstein, Trinomial Triangle From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring Hockey Stick Theorems: Proof of Results and References

Author:
(1) Sima Mehri, Farzanegan High School. 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) 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. Author: (1) Sima Mehri, Farzanegan High School. Author: Author: (1) Sima Mehri, Farzanegan High School. Table of Links Abstract and 1 Introduction and Description of Results Abstract and 1 Introduction and Description of Results 2. Proof of Results and References 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) 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) 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) 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. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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