REPDIGITS IN THE BASE b AS SUMS OF FOUR BALANCING NUMBERS

The sequence of balancing numbers (B-n) is defined by the recurrence relation B-n = 6B(n-1) - Bn-2 for n >= 2 with initial conditions B-0 = 0 and B-1 = 1. B-n is called the nth balancing number. In this paper, we find all repdigits in the base b, which are sums of four balancing numbers. As a result of our theorem, we state that if B-n is repdigit in the base b and has at least two digits, then (n, b) = (2, 5), (3, 6). Namely, B-2 = 6 = (11)(5) and B-3 = 35 = (55)(6).