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

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