Is there any reason why neutrons and protons can't combine in a P-N-N-N-P type chain rather than in a tightly packed sphere?
A chain would be highly unstable, the Strong Nuclear Force would immediately pull the protons and neutrons together, to minimise their potential energy.
A chain would also distort the spherical symmetry of the electrostatic force, the orbital patterns and energy levels of electrons would be completely different than we observe.
The existence of highly stable shells in nuclei have given rise to observations of molecule-like arrangements in nuclei. In particular in 12-Be (8 neutrons, 4 protons) the arrangement of the nucleons is often better described as two highly stable alpha particles (two protons, two neutrons) bound by the four extra neutrons.
However, although nuclei are able to form exotic structures like this at neutron, proton numbers far from the closed shells, this is not due to the single inter nucleon forces (as your "P-N-N-N-P type chain" might allude to) but the collective effects of grouped nucleons and the potentials they create.
A rapidly spinning nucleus can form into a linear chain of several small clusters of neutrons and protons. See http://physics.aps.org/story/v28/st10. This is not a P-N-N-N-P type chain as per your question, but still very far from a sphere.