For odd $v \geq 5$, Schmerl and Spiegel have proved that the
$1$-additive sequence $(2,v)$ has precisely two even terms and,
consequently, is regular. For $5 \leq v \equiv 1 \bmod{4}$, 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 $\Delta (v)$ of $(4,v)$ tends to $0$
as $v \to \infty$, but is misbehaved in the sense that
¶ $$
\eqalign{
\liminf_{\textstyle{\vrule height 5pt width 0pt
v \to \infty\atop v \equiv 1 \bmod{4}}}
\left(\frac{v}{2}\right)^{2-\log_23}\Delta (v)=\quarter,\cr
\limsup_{\textstyle{v \to \infty\atop v \equiv 1 \bmod{4}}}
\left(\frac{v}{2}\right)^{2-\log_23}\Delta (v)>
0.27164.}
$$
¶ This is proved using techniques adapted from Harborth and Stolarsky.