Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1982. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 16)

II. TANULMÁNYOK A TERMÉSZETTUDOMÁNYOK KÖRÉBÖL - Dr. Mátyás Ferenc: Wythoff-párok és a másodrendű sorozatok kapcsolata

SUMMARY WYTIIOFF-PAIRS AND SECOND ORDER RECURRENCES by Ferenc Mátyás In this paper we deal with the connection between second order linear recurrences and Wythoff-pairs. Let G — G {A, B. G

0 G

t) =~{G„}®

o be a second order linear recurrence defined by integer constants A. B, G

0, Gt and the recurrence G

n = AG

n_i+BG

n_

2 (n > 1), where A B =(= 0, D = A

2 + 4B 4= 0 and [G

0| + |G,| 4= 0. If a and /? are the roots of the equation x

2 — Ax — B = 0, then we have G„ = a «

n — b /3

n, where Gi —B Go Gi — a Go a

= a

nd a — p a — p Let us define the sequences {u„}

n=

1 and {t>

n}

n=i in the following manner: Ui : == 1, v

L : = 2 and for k > 1 u

k : — m, where m is the smallest positive integer for which u

i 4= m, vj =t= m if 1 = i < k and v

k : = u

k + k. The pairs (uj.vi), (U2; V2), ... are called Wythoff-pairs. We prove the following theorem

: Let D > 0. The equation (G

n; G

n+

k) = (

U

i; v

t) has an iinfanite number of solutions in integer n. k, i, iff A = B —k=l and a > 0. 555