Author:
(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

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

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Exploring Hockey Stick Theorems: Proof of Results and References

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