  Euler's continuants are universal polynomials expressing the numerator and
denominator of a finite continued fraction whose entries are independent
variables. We introduce their categorical lifts which are natural complexes
(more precisely, coherently commutative cubes) of functors involving
compositions of a given functor and its adjoints of various orders, with the
differentials built out of units and counits of the adjunctions. In the stable
infinity-categorical context these complexes/cubes can be assigned
totalizations which are new functors serving as higher analogs of the spherical
twist and cotwist. We define N-spherical functors by vanishing of the twist and
cotwist of order N-1 in which case those of order N-2 are equivalences. The
usual concept of a spherical functor corresponds to N=4. We characterize
N-periodic semi-orthogonal decompositions of triangulated (stable infinity-)
categories in terms of N-sphericity of their gluing functors. The procedure of
forming iterated orthogonals turns out to be analogous to the procedure of
forming a continued fraction.
