  We prove a conjecture of Northshield by determining the maximal order of his
analogue of Stern's sequence for $\mathbb{Z}[\sqrt{2}]$. In particular, if $b$
is Northshield's analogue, we prove that
$$\limsup_{n\to\infty}\frac{2b(n)}{(2n)^{\log_3 (\sqrt{2}+1)}}=1.$$
