Cycles of higher-order Collatz sequences

Abstract

Consider a sequence of numbers x(n) is an element of Z(+) defined by x(n+1) = x(n)/2 if x(n) is even, and x(n+1) = x(n) + 2x(n-1) + q/2 if xn is odd. A 1-cycle is a periodic sequence with one transition from odd to even numbers. We prove theoretical and computational results for the existence of 1-cycles, and discuss a generalization to more complex cycles.