On the residue classes of integer sequences satisfying a linear three term recurrence formula

Abstract

In this paper we investigate linear three-term recurrence formulae Z n = T ( n ) Z n - 1 + U ( n ) Z n - 2 ( n ⩾ 2 ) with sequences of integers ( T ( n ) ) n ⩾ 0 and ( U ( n ) ) n ⩾ 0 , which are ultimately periodic modulo m, e.g. ( T ( n ) mod m ) n ⩾ 0 = ( a 0 , a 1 , a 2 , … , a ρ , T 1 , T 2 , … , T w ¯ ) ( U ( n ) mod m ) n ⩾ 0 = ( b 0 , b 1 , b 2 , … , b ρ , U 1 , U 2 , … , U w ¯ ) . In a former paper of this journal the authors computed explicitly the coefficients of a linear three-term recurrence formula for z n = Z rn + i with 0 ⩽ i < r , when ( T ( n ) ) n ⩾ 0 and ( U ( n ) ) n ⩾ 0 belong to regular or non-regular Hurwitz-type continued fraction expansions. Using this result we show now that the sequence ( Z n ) n ⩾ 0 is ultimately periodic modulo m. As a consequence, for Hurwitz-type continued fraction expansions α = [ a 0 ; T 1 ( k ) , … , T r ( k ) ¯ ] k = 1 ∞ or α = [ a 0 ; a 1 , T 1 ( k ) , … , T r ( k ) ¯ ] k = 1 ∞ with polynomials T 1 ≠ const . , T 2 , … , T r we deduce for all positive integers a and m that liminf q q ‖ q α ‖ = 0 , where q ≡ a mod m and ‖ · ‖ denotes the distance from the nearest integer. Finally, we are particularly interested in the recurrence formula Z n = ( an + b ) Z n - 1 + cZ n - 2 ( n ⩾ 2 ) and compute the length of a period of the sequence ( Z n mod m ) n ⩾ 0 , when ( a , m ) divides b, and ( c , m ) = 1 . This generalizes former results of the authors dealing with regular continued fraction expansions of the numbers exp ( 1 / s ) for integers s ⩾ 1 .