  In a recent letter we presented the equations which describe tensionless
limit of the excited-state spectrum for strings on $AdS_3\times S^3\times T^4$
supported by Ramond-Ramond flux, and their numerical solution. In this paper,
we give a detailed account of the derivation of these equations from the mirror
TBA equations proposed by Frolov and Sfondrini, discussing the
contour-deformation trick which we used to obtain excited-state equations and
the tensionless limit. We also comment at length on the algorithm for the
numerical solution of the equations in the tensionless limit, and present a
number of explicit numerical results, as well as comment on their
interpretation.
