Fourth Order Linearly Recurrent Wythoff Pairs

Abstract

An examination of the sizable literature on Wythoff pairs, their generalizations, and Beatty sequences shows a considerable emphasis on connections with second order recurrence relations and quadratic irrationalities. However, it does not seem to have been previously noticed that a subsequence of the classical Wythoff pairs satisfies the irreducible fourth order linear recurrence $$C_{n + 4} = 10C_{n + 3} - 16C_{n + 2} + 5C_{n + 1} + C_n . $$ Thus, while Wythoff pairs are ordinarily associated with the roots of x2 − x − 1 = 0, there is a subset thereof that is associated with the roots of $$ x^4 - 10x^3 + 16x^2 - 5x - 1 = 0. $$