Starts at: 2025-03-01 10:45AM
Ends at: 2025-03-01 12:00PM
Abstract:
We introduce a triangle of numbers, similar to the Fibonacci Triangle and the Pascal Triangle and discuss some of its properties. As an open question, this triangle was given as an OEIS Challenge (P419) from “The Playground” in the Math Horizons magazine. The triangle is constructed in the following way: The entries of the first diagonal from the left (or the first back slash diagonal) of the triangle are all equal to 2, and the second back slash diagonal is made up of the Fibonacci numbers. Any entry in a particular back slash diagonal after that is obtained using a recursive formula that involves adding up all previous entries in the respective diagonal along with a constant which depends on the position of the entry that is being computed. We provide a formal definition for the entries of the triangle using a recursive formula. We also discuss how an L-shaped pattern in the triangle gives rise to a formula for the entries of the third slash diagonal (or third diagonal from the right), involving entries of the previous slash diagonals and Triangular numbers. We finally provide a generalized proof of this formula using Mathematical Induction. Other patterns are also included.