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On the Recurrence F<sub>m+1</sub> = B<sub>m</sub>f<sub>m</sub>-F<sub>m-1</sub> and Applications
BibTeX
@MISC{Fraenkel_onthe,
author = {A.S. Fraenkel and Aviezri S. Fraenkel and Z},
title = {On the Recurrence F<sub>m+1</sub> = B<sub>m</sub>f<sub>m</sub>-F<sub>m-1</sub> and Applications},
year = {}
}
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Abstract
. We point out that a natural and direct way of analyzing the recurrence fm+1 = b m fm \Gamma fm\Gamma1 is by means of continued fractions theory. Applications to exotic numeration systems and to games are given, and previous examinations of the special case fm+1 = 6fm \Gamma fm\Gamma1 are pointed out. Denote by Z 0 and Z + the set of nonnegative integers and positive integers respectively. We denote simple continued fractions in the form (1) ff = a 0 + 1 a 1 + 1 a 2 + 1 . . . := [a 0 ; a 1 ; a 2 ; : : : ] ; and its convergents by pn =q n = [a 0 ; : : : ; an ] (n 0). Putting p \Gamma1 = 1, q \Gamma1 = 0, p 0 = a 0 , q 0 = 1, we then have pn = an pn\Gamma1 + pn\Gamma2 , q n = an q n\Gamma1 + q n\Gamma2 (n 1). See e.g., [HaWr1989, Ch. 10], [Per1950]. All the partial quotients a i are positive integers. Let un stand for either pn or q n , with the understanding that in each formula, either all u i denote p i or all denote q i . c fl0000 American Mathematical Society ???-????/00...
Keyphrases
sub sub sub sub sub m-1 sub positive integer direct way exotic numeration system gamma1 gamma2 nonnegative integer partial quotient fm gamma fm gamma1 fl0000 american mathematical society pn gamma1 pn gamma2 simple continued fraction special case previous examination continued fraction theory gamma fm gamma1
