  Exact solvability (typically, of harmonic oscillators) in quantum mechanics
usually implies an elementary form of the spectrum while in all the
"next-to-solvable" models, the energies E are only available in an implicit
form (typically, as eigenvalues of an N-dimensional matrix). We demonstrate
here that certain echoes of the unattainable harmonic-oscillator ideal may
still survive in the latter (often called quasi-exact) cases exemplified here
by the popular sextic anharmonic oscillator. In particular we show that
whenever the spatial dimension D (or, equivalently, angular momentum L) happens
to be "sufficiently" large, the surprisingly compact semi-explicit energies E
remain available. In detail, using the Rayleigh-Schr\"{o}dinger perturbation
theory in its appropriate "shifted-L" version we observe that: (1) all the k-th
order approximants $E_k$ remain defined in integer arithmetics (i.e., without
any errors); (2) an optimal auxiliary N-dependent shift $\beta(N)$ of L exists
and is unique; (3) the resulting perturbative E degenerates to the series in
powers of $1/[L+\beta(N)]^2$; (4) a certain optimal Pade re-summation formulae
exist and possess a generic branched-continued-fraction structure.
