A Class of 1-Additive Sequences and Quadratic Recurrences

Julien Cassaigne Steven R. Finch

Abstract

For odd v ≥ 5, Schmerl and Spiegel have proved that the 1-additive sequence (2, v) has precisely two even terms and, consequently, is regular. For 5 ≤ v ≡ 1 mod 4, we prove, using a different approach, that the 1-additive sequence (4, v) has precisely three even terms. The proof draws upon the periodicity properties of a certain ternary quadratic recurrence.

Unlike the case of (2, v), the regularity of (4, v) can be captured by expressions in closed form. For example, its period can be written as an exponential sum of binary digit sums. Therefore the asymptotic density Δ(v) of (4, v) tends to 0 as v → ∞, but is misbehaved in the sense that

lim inf_{{v→∞ \atop v≡1mod4}} (\frac{v}{2} )^{2−log_{2} 3} Δ(v) = \frac{1}{4},

lim sup_{{v→∞ \atop v≡1mod4}} (\frac{v}{2} )^{2−log_{2} 3} Δ(v) > 0.27164.

This is proved using techniques adapted from Harborth and Stolarsky.