ON k-GENERALIZED FIBONACCI NUMBERS WITH ONLY ONE DISTINCT DIGIT

Let (F-n)(n >= 0) be the Fibonacci sequence, defined by the recurrence Fn+2 = Fn+1+F-n, with initial values F-0 = 0 and F-1 = 1. In 2000, Florian Luca proved that F-10 = 55 is the largest number with only one distinct digit (repdigit) in the Fibonacci sequence. The Tribonacci numbers (T-n)(n >= 0) are like the Fibonacci numbers, but starting with 0,0, 1, and each term afterwards is the sum of the preceding three terms. In this paper, we prove that 44 is the largest Tribonacci number with only one distinct digit. Actually, we shall use transcendental tools to provide a general method for searching repdigits in a k-generalized Fibonacci sequence and we also make a conjecture based on some cases.