  An interesting aspect of antiferromagnetic quantum spin ladders, with
complete dimer coverings, is that the wave function can be recursively
generated by estimating the number of coverings in the valence bond basis,
which follow the fabled Fibonacci sequence. In this work, we derive generalized
forms of this sequence for multi-legged and doped quantum spin ladders, which
allow the corresponding dimer-covered state to be recursively generated. We
show that these sequences allow for estimation of physically and
information-theoretically relevant quantities in large spin lattices without
resorting to complex numerical methods. We apply the formalism to calculate the
valence bond entanglement entropy, which is an important figure of merit for
studying cooperative phenomena in quantum spin systems with SU(2) symmetry. We
show that introduction of doping may mitigate, within the quarters of
entanglement entropy, the dichotomy between odd- and even- legged quantum spin
ladders.
