Fibonacci structures related to adjoint functors and semi-orthogonal decompositions

Various contractions of the composition of a string of iterated adjoint functors form
a diagram whose shape is known in computer science as the "Fibonacci cube".
Such diagrams can be seen as categorical lifts of Euler continuants, the universal numerators
and denominators of finite continued fractions whose entries are independent variables.
Vanishing (exactness) of the totalization of the (N-1)st Fibonacci cube defines the concept
of an N-spherical functor, with usual spherical functors appearing
for N=4. They are related to N-periodic semi-orthogonal decompositions in triangulated categories.
Joint work in progress with T. Dyckerhoff and V. Schechtman.