Initial number of Lucas’ type series for the generalized Fibonacci sequence

Abstract

Initial numbers for Lucas? type series have so far been established only for Fibonacci (2,1) and Tribonacci (3,1,3) sequences. Characteristics of stated series is their asymptotic relation with the exponent of the series constant. By using a simple procedure based on asymptotic relations of exponents of a sequences constant and Lucas? type series with the application of Nearest Integer Function - NIF, a general rule for initial numbers of Lucas? type series of Generalized Fibonacci sequence has been established, for the first time. All the gained initial numbers are integers, first initial number is always equal to the order of the sequence Fn(0) = n and remaining are functionally dependent on order of the number and are equal to Fn(k) = 2k-1-1. This is premiere presentation of Prim-nacci sequence, too. Determinants of initial numbers of the Lucas? type series for the generalized Fibonacci sequences are a proven factorial function.