Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)
LlPTAI, K., An approximation problem concerning linear recurrences
12 Kálmán Liptai Horadam [2] showed that the points P n = (R n , R n+i ) he on the conic section Bx 2 - Axy + y 2 + eB n = 0, where e = AR 0R\ - BR 2 - R\ and the intial terms RQ and R\ are not necessarily 0 and 1. For the Fibonacci sequence, when A — 1 and B = — 1, C. Kimberling [6] characterized conics satisfied by infinetely many Fibonacci lattice points (x,y ) = (F m,F n ). J. P. Jones and P. Kiss [4] considered the distance of points P n = (R n, R n+i) and the line y = ax. They proved that this distance tends to zero if and only if \ß\ < 1. Moreover, they showed that in the case \B\ = 1 there is not such a lattice point (x,y) which is nearer to the mentioned line than P n, if |x| < |Ä n|. They proved similar arguments in three-dimensional case, too. In this paper we investigate the geometric properties of the lattice points P n = (R n, Rn+i , V n). We shall use the following result of P. Kiss [5]: if = 1 and p/ q is a rational number such that (p, q) = 1, then the inequality P a q < VDq< implies that v/ q = R n+i/R n for some n > 1. It is known, that (3) lim = a n-> oo K n and (4) lim ^r = >/D n—>oo K n (see. e.g. [3], [7]). Let us consider the vectors (R n, R n+i , V n). Since by (3) and (4) and using the equality (R n, Rn+l , Vn) ~ Rn (1, ^ , ) V -fin TIN J we get that the direction of vectors (R n, R n+i , Vn) tends to the direction of the vector a, y/D^j . However, the sequence of the lattice points P n = (R n, R n+i , V n) does not always tend to the line passing through the origin and parallel to the vector Vd), we give a condition when it is hold. Theorem 1. Let L be the line defined by x = t, y = at , z — y/Dt , t E R. Futhermore, let d n be the distance from the point (R n, R n+i , V n) (n = 0,1, 2,...) to the line L. Then lim d n = 0 if and only if \ß\ < 1.
