Boron is an element that has ability to build strong and highly directional bonds with boron itself. As a result, boron atoms form diverse structural motifs, ultimately can yield distinct nano structures, such as planar, quasi-planar, convex, cage, open-cage, tubular, spherical, ring, dome-like, shell, capsule, and so on, i.e., it can take almost any shape. Therefore, a deep understanding of the physical and chemical properties becomes important in boron cluster chemistry. Electronic and geometric structures, total and binding energies, harmonic frequencies, point symmetries, charge distributions, dipole moments, chemical bondings and the highest occupied molecular orbital-owest unoccupied molecular orbital energy gaps of neutral Bn (n=13～20) clusters have been investigated by density functional theory (DFT),B3LYP with 6-311++G(d,p) basis set. Furthermore, the first and the second energy differences are used to obtain the most stable sizes. We have observed that almost all physical properties are size dependent, and double-ring tubular form of B20 has the highest binding energy per atom. The icosahedral structure with an inside atom is found as impossible as a stable structure for the size thirteen. This structure transforms to an open-cage form. The structural transition from two-dimensional to three-dimensional is found at the size of 20 and consistent with the literature. The calculated charges by the Mulliken analysis show that there is a symmetry pattern with respect to the x-z and y-z planes for the charge distributions. The unusual planar stability of the boron clusters may be explained by the delocalized Π and α bonding characteristic together with the existence of the multicentered bonding. The results have been compared to available studies in the literature.