The third order Nickel Fibonacci numbers

Abstract Using the technique of characteristic equations and De Moivre-type identity, this article defines the third order Nickel Fibonacci sequence, the third order Nickel Lucas and a modified third order Nickel Fibonacci sequence. We obtain some well-known identities for the sequences such as the Binet’s identity, Cassini identity, generating functions, and matrix representations. Also, we present D’Ocagne’s identity and Melham’s identity. We develop and illustrate a new encryption/decryption algorithms in which the keys are the Nickel Fibonacci matrices. Moreover, we give MATLAB-R2017a codes for encryption and decryption algorithms.