ON PERIODICITY OF RECURRENT SEQUENCES OF THE SECOND AND THE THIRD ORDER

Abstract

Among other sequences of integers Fibonacci numbers and Lucas numbers are cituated in the central place. In spite of great amount of literature dedicated to Fibonacci and Lucas sequences, there are still a lot of intriguing questions and open problems in this direction, see, for instance, the ''The Fibonacci Quarterly'' journal or materials of the Biannual International Conference organized by Fibonacci Association.Among other sequences of integers Fibonacci numbers and Lucas numbers are cituated in the central place. In spite of great amount of literature dedicated to Fibonacci and Lucas sequences, there are still a lot of intriguing questions and open problems in this direction, see, for instance, the ''The Fibonacci Quarterly'' journal or materials of the Biannual International Conference organized by Fibonacci Association. We are motivated by the following simple observatoin. Consider the classical Fibonacci sequence defined by the rule $$ F_{n+2}=F_{n+1}+F_n, n=0,1,2,\dots $$ with the initial values $F_0=0$, $F_1=1$: $$ 0,1,1,2,3,5, 8, 13, 21, 34, 55,\dots $$ If we consider a little bit another sequence $$ G_{n+2}=G_{n+1}-G_n, n=0,1,2,\dots, $$ then for $G_0=0$, $G_1=1$ the sequence $(G_n)_{n=0}^\infty$ is of the form $$ 0,1,1,0,-1,-1,0,1,1,0,-1,-1,\dots. $$ In other words, this sequence is periodic with period of the length $6$. Therefore, the next  questions   follow naturally from the previous observation:(i) under which conditions on its coefficients the reccurent sequence is periodic? (ii) How long may be a period of the reccurent sequence and how it depends on coefficients? (iii) Does the length of a period depends on initial values of the reccurent sequence?   In the given paper we answer  to these questions for the reccurent sequences of the second and the third order. We obtain necessary and sufficient conditions on coefficients $u_i$ for the periodicity of a recurrent sequence defined by the rule  $a_{n+k}=u_{k-1}a_{n+k-1}+\dots+u_0a_0$ for $n=0,1,\dots$ and $u_i\in\mathbb R$, $i=0,\dots,k-1$, in the case of $k=2,3$.