Iterated rascal triangles

Authors

Jena M. Gregory, The University of Texas Rio Grande Valley
Brandt Kronholm, The University of Texas Rio Grande Valley
Jacob White, Texas A &M University

Document Type

Article

Publication Date

8-2024

Abstract

In this paper we consider a number triangle known as the Rascal triangle and place it within an infinite sequence of number triangles we call Iterated Rascal triangles. Initially, the name “Rascal triangle” was a clever play of words. However, we show a surprising and direct connection between Iterated Rascal triangles and Pascal’s triangle. We prove an unexpected and elegant recursive formula generates the entries in any Iterated Rascal triangle. We establish formulas, generating functions, and combinatorial identities for them. From these principals we prove a version of Lucas’ Theorem for the Iterated Rascal triangles. Moreover, we consider an identity from the Rascal triangle that has a well-known analogous identity in Pascal’s triangle and extend it to all Iterated Rascal triangles. Although many can be made, we conclude with one conjecture.

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Recommended Citation

Gregory, Jena, Brandt Kronholm, and Jacob White. "Iterated rascal triangles." Aequationes mathematicae 98, no. 4 (2024): 1115-1132. https://doi.org/10.1007/s00010-023-00987-6

Publication Title

Aequationes mathematicae

DOI

10.1007/s00010-023-00987-6