Generalizations and applications of Padovan numbers

In this study, various generalizations and application areas of the Padovan number sequence, which has an important place among the number sequences, are examined. It is known that the Padovan number sequence has a mathematical relationship with the plastic constant used in architecture. In this thesis, firstly, the geometrical shapes of the plastic constant and its various applications are presented. Then, the generalized Padovan polynomial sequence is defined and the general solution of this sequence depending on the roots of the characteristic equation, generator functions, matrix representation and some identities are examined. Then, weighted Padovan sum formulas are obtained by means of the Padovan sum formula. The study continues as follows: • Fibonacci-Padovan sequence, which is a relation related to Fibonacci and Padovan numbers, • Bi-periodic Padovan sequence in the form of a piecewise function that gives different results in the odd and even case of the indices of the Padovan sequence, • Symmetrical Padovan sequence established with row and column relationship, • Pulsating Padovan number sequence consisting of pair sequences with a different sequence of numbers, technically also called the pulse beat, • Padovan triangle sequence resembling Pascal's triangle obtained by using Padovan numbers is defined and various identities are included. In addition to these generalizations, Bernoulli Padovan numbers and polynomials are investigated and different results are obtained by using new definitions and various methods. Finally, generalized Padovan block polynomial matrices are defined using generalized Padovan polynomial matrices, and some properties are investigated. A key agreement method is given for key exchange, and a public key cryptography is proposed, inspired by the extended Hill encryption and affine encryption system, and the reliability of the system is analyzed.