Search

In this paper we study the b-ary expansions of the square roots of the function defined by the recurrence ƒb(n) = b ƒb(n−1) + n with initial value ƒ(0) = 0 taken at odd positive integers n, of which the special case b = 10 is often referred to as the ''schizophrenic'' or ''mock-rational'' numbers. Defined by Darling in 2004 and studied in more detail by Brown in 2009, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 by formally defining the schizophrenic pattern present in the b-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within. [ABSTRACT FROM AUTHOR]