Research On The Identities For The Generalized 3-periodic Fibonacci Sequences

The research of Fibonacci sequences have high value and are widely used in many fields.In mathematics,calculating the limit of the quotient of two adjacent terms by the general formula of Fibonacci sequences,we obtain the golden ratio.Many useful identities can be obtained by studying the properties of Fibonacci sequences.In addition,we can study the Fibonacci sequences to understand its own congruence properties.In this paper,we consider the generalized 3-periodic Fibonacci sequences.In this paper,we study some kinds of identities of the generalized 3-periodic Fibonacci sequences.Firstly,we construct the general formula of the generalized 3-periodic Fibonacci sequences,then under certain conditions,we give some identities of the sum of binomial coefficients for the generalized 3-periodic Fibonacci sequences and the generalized 3-periodic Lucas sequences by matrix method.Secondly,we give the relationships between the generalized 3-periodic Fibonacci sequences and the generalized 3-periodic Horadam sequences.Finally,by constructing the matrix and using matrix method and algebraic manipulation,we derive the second-order and fourth-order identities of the generalized 3-periodic Horadam sequences.As an application,we derive a congruence property satisfed by the generalized3-periodic Horadam sequences.