Exploring Hockey Stick Theorems: Proof of Results and References

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Investigating hockey stick theorems in Pascal's and trinomial triangles, revealing patterns akin to ball trajectories in combinatorial mathematics.
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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.