Quasi-polynomial continued-fraction conjecture for $g(N)=\sqrt{R_N/2}$

Let $R_N$ be the quantity defined in the paper, let $g(N)=\sqrt{R_N/2}$, and let $N_k$ and $Q_{2k+1}$ be the sequences defined above. Write $\operatorname{PQ}_j(g)$ for the $j$-th partial quotient in the continued fraction of $g$. The formal continued fraction of $g(N)$ is

$$g(N)=\Bigl[0;\,2N+1,\,4N+2,\,\Bigl\lfloor\frac{12N+6}{19}\Bigr\rfloor,\,\ldots\Bigr],$$

with quasi-polynomial partial quotients of periods $1,1,19,\ldots$. In particular, at $N=N_k$, the quasi-polynomial continued-fraction conjecture.

$$\operatorname{PQ}_1(g)\big|_{N_k}=2N_k+1=Q_{2k+1},\qquad \operatorname{PQ}_2(g)\big|_{N_k}=4N_k+2=2Q_{2k+1}.$$

This pattern is suggested by the formal Euler–Maclaurin and Euclidean continued-fraction calculations. Unlike the six partial quotients established in the paper's main theorem, it is not rigorously established at the smallest indices; the corresponding formal statement remains an open problem in the accompanying Lean development.