Fractal Characteristics of Horadam Sequences with Positive Discriminants

We study the fractal characteristics of Horadam sequences and generalize a recent result on the Fibonacci sequence to Horadam sequences with positive discriminants. More precisely, for any integers [Formula: see text] with [Formula: see text], we prove that the Horadam sequence [Formula: see text] defined by [Formula: see text] [Formula: see text] and [Formula: see text] for all [Formula: see text] satisfies [Formula: see text] where [Formula: see text] denotes the box dimension. The main tool we use is the Binet formula, together with box-dimension estimates. The hypothesis [Formula: see text] is sharp.