Some Identities on Generalized Arithmetic Triangles

Summary

We generalize the arithmetic triangle of Blaise Pascal, Yang Hui, and others, by maintaining its recurrence relation, but replacing the traditional 1s on the boundary with arbitrary sequences. Within these structures, we examine some identities commonly studied in Pascal’s version, including those related to sums and alternating sums of entries in rows as well as the so-called hockey stick identities. We use recurrence relations and elementary generating functions to derive general results and see how these results can be used in some interesting special cases.

MSC:

• 05A10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Douglas E. Ensley

Doug Ensley (MR Author ID: 608045) holds mathematics degrees from the University of Alabama, Huntsville and Carnegie Mellon University. He recently retired from teaching after 29 years at Shippensburg University and currently works part-time as Director of the MAA OPEN Math program from his home near Wilmington, NC.

Ji Young Choi

Ji Young Choi (MR Author ID: 690769) is a professor of mathematics at Shippensburg University of Pennsylvania. She earned her BS and MS in mathematics at Busan National University in Korea, and her Ph.D. in mathematics at Iowa State University. Her main areas of research are enumerative combinatorics and elementary number theory. She likes to work on integer sequences, and she enjoys writing and working on mathematics competition problems for elementary, middle, and high school students.

Jesica Hoover

Jesica Hoover is an undergraduate student in the Wood Honors College at Shippensburg University and a mathematics major pursuing secondary-level teaching certification. Her work on this project was supported by the Shippensburg University Summer Undergraduate Research Experience program.