  Motivated by a recent experiment [L.Chomaz et al., Nature Physics 14, 442
(2018)], we perform numerical simulations of a dipolar Bose-Einstein Condensate
(BEC) in a tubular confinement at T=0 within Density Functional Theory, where
the beyond-mean-field correction to the ground state energy is included in the
Local Density Approximation. We study the excitation spectrum of the system by
solving the corresponding Bogoliubov-de Gennes equations. The calculated
spectrum shows a roton minimum, and the roton gap decreases by reducing the
effective scattering length. As the roton gap disappears, the system
spontaneously develops in its ground-state a periodic, linear structure formed
by denser clusters of atomic dipoles immersed in a dilute superfluid
background. This structure shows the hallmarks of a supersolid system, i.e. (i)
a finite non-classical translational inertia along the tube axis and (ii) the
appearance, besides the phonon mode, of the Nambu-Goldstone gapless mode
corresponding to phase fluctuations, and related to the spontaneous breaking of
the gauge symmetry. A further decrease in the scattering length eventually
leads to the formation of a periodic linear array of self-bound droplets.
