In this paper, we define and investigate modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo sequences as special cases of the generalized Leonardo sequence. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there are close relations between modified p-Leonardo, p-Leonardo-Lucas, p-Leonardo numbers and Fibonacci, Lucas numbers.

Leonardo da Vinci (1452-1519) is considered one of the greatest geniuses of the Renaissance. His studies developed advanced ideas for his time, even in its most unknown aspects, such as: architecture, urban planning and restoration. He never studied formally, but he learned everything due to his method of observation, the study of other treatises and especially the group of artists with whom he collaborated. After the study of its codices a great interest and knowledge is detected, related to the design, the structural calculation, the materials and the constructive systems, in such a way that their proposals influenced the architecture of the Renaissance, through the work of other authors of your time. The purpose of this article is to make a critical analysis of its excellent architectural proposal in the cupola of the Duomo of Milan and the reasons why it was not carried out under its name. This proposal is included within a discipline in which it has never been recognized as such, but which demonstrated the same qualities as in other fields where it is recognized prestigiously.

Among many other themes, Leonardo da Vinci’s Manuscript B contains several drawings of centrally planned churches, some of which are represented using a plan view paired with a bird’s-eye view. The use of a bird’s-eye instead of an elevation represents an innovative depiction technique, which allows to combine the immediacy of the perspective view with the measurability of the façades, and therefore to describe the three-dimensionality of the buildings. To understand the reasons behind the use of this original technique, the edifices’ shape and classify them we decided to use 3D digital reconstruction techniques, for their ability to avoid misunderstandings in the reconstruction process and in the results. This article describes the method to create the digital models of sixteen churches. A Visual programming language (VPL) script was used as 3D base for modelling the churches achieved from a classification code expressing the aggregative rules of the churches. Then, the geometric process for the construction of the plan and its relationship with the elevation measures was studied for each church. Finally, this information was used for the completion of the 3D models, distinguishing more output variants each time there was an inconsistency between plan and perspective view, a variability of one architectural element or an uncertainty.