Exploring Hockey Stick Theorems: Proof of Results and Referencesby@hockeystick

# Exploring Hockey Stick Theorems: Proof of Results and References

June 26th, 2024

In the proof of both theorems, we use induction. This paper is available on arxiv(https://arxiv.org/abs/1404.5106) under CC BY 4.0 DEED license. The hockey stick theorem in the trinomial triangles has been proved. It can be translated in Pascal pyramid as follows.

Author:

(1) Sima Mehri, Farzanegan High School.

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.

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