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Volume 1 | 2018
We would like to thank The Kalhert Foundation for their support of our students and specifically for this journal. Defining Katie’s Triangle and Common Properties with Pascal’s Triangle Charles Lundelius, Howard Community College Jake Hoffman, Howard Community College Mentored by: Mike Long, Ed.D & Loretta FitzGerald Tokoly, Ph.D.
Abstract
A triangular array, known as Katie’s Triangle, was derived from Pascal’s Triangle. The talking about it in the abstractarray was initially derived directly using the elements in Pascal’s Triangle, but later a general form that uses an iterative process with the elements of Katie’s Triangle was derived. For any element in Katie’s Triangle, a term in the n + 1 row takes the larger numerator and denominator of the adjacent upper left and upper right terms of the nth row. Once a derivation for Katie’s Triangle was found, the structure of it was studied and compared to that of Pascal’s Triangle to determine if there were similar properties. Some of the properties were found to exist in both: including the flowers and the Triangular Numbers. While the flowers property held exactly the same for both triangles, the Triangular Numbers existed albeit in a different arrangement than in Pascal’s Triangle. Other properties were found to either not exist at all, or were not one-to-one matches with Pascal’s properties. These include sum of the rows, sum of the first n Figurate Numbers, and the existence of most figurate numbers. Despite being derived initially from Pascal’s Triangle, Katie’s Triangle stands on its own.
Paper
A colleague, Katie, was analyzing Pascal’s Triangle when she decided to multiply each element to get the next one in a row. The resulting needed factors were then removed and formed into their own triangle. From here, the new triangle was analyzed for properties similar to Pascal’s Triangle. After generating Katie’s Triangle, the question arose as what to relationships, if any, were translated from Pascal’s Triangle to this new triangle. A second question was if there were new properties that existed in Katie’s Triangle. Katie’s Triangle is constructed starting with at the peak. Each successive row has one more entry than the previous row. The numerators start at one and increase by one moving across the row from right to left while the
Volume 1 | 2018 44 denominators start at one and increase by one moving from left to right. Any term in the triangle can be expressed by the following general formula with the peak defined as row zero and diagonal zero: