Generalization of the Fibonacci sequence, Pascal's triangle, and the binomial theorem

Abstract

This research generalizes the relationship between Pascal's triangle and binomial expansion by using variables instead of numbers. The triangle is formed using variable (d) instead of the zero term (0), variable (a) instead of the first term (1), and variable (m) as a generalization of the binomial theorem. The mathematical patterns resulting from the triangle's formation are studied using these variables, leading to five new mathematical equations: the vertical equation, the hypotenuse equation, the row equation, the sum of the rows equation, and the equation of the golden function sequences. The equation of the golden function sequences is considred an unprecedented generalization of the nth term of the Fibonacci and Lucas sequence. Additionally, a new, unprecedented diamond equation is for mulated, and a new conjecture related to prime numbers is formulated, as it is considered a generalization of Fermat's Little Theorem. This research highlights the need for a more comprehensive understanding of the relationship between Pascal's triangle and binomial expansion.