On the Periods of Generalized Fibonacci Recurrences

On the periods of generalized Fibonacci recurrences

We give a simple condition for a linear recurrence (mod 2W) of degree r to have the maximal possible period 2M_1(2'' 1). It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e., for three-term linear recurrences defined by trinomials which are primitive (mod 2) and of degree r > 2 . We consider the enumeration of certain exceptional polynomials wh...

On the Periods of Generalized Fibonacci Recurrences1

We give a simple condition for a linear recurrence (mod 2w) of degree r to have themaximal possible period 2w 1(2r 1). It follows that the period is maximal in the casesof interest for pseudo-random number generation, i.e. for 3-term linear recurrences de nedby trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumera-tion of certain exceptional pol...

Generalized Fibonacci recurrences and the lex-least De Bruijn sequence

The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographicallyleast de Bruijn sequence at natural breakpoints of the binary string. We show that the skew and length of these suffixes are enumerated by sequences generalizing the Fibonacci and L...

G. Sburlati GENERALIZED FIBONACCI SEQUENCES AND LINEAR RECURRENCES

We analyze the existing relations among particular classes of generalized Fibonacci sequences, with characteristic polynomial having two integer roots. Several examples are given. Some properties concerning in general linear recurrences are also shown.

On the Periods of 2-Step General Fibonacci Sequences in the Generalized Quaternion Groups