On the recurrence fm+1=bmfm-fm-1 and applications

There's a proposition proclaimed to be new; it existed in eternity before us. The early authors are forgotten; and also the contemporary ones won't have any remembrance, though they are modern. Abstract We point out that a natural and direct way of analyzing the recurrence fm+1 = bmfm ?fm?1 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 ? fm?1 are pointed out. Denote by Z, Z 0 and Z + the set of integers, nonnegative integers and positive integers respectively. If a 0 2 Z, a i 2 Z + (i 1), then = a 0 + 1 a 1 + 1 a 2 + 1. .. Let u n stand for either p n or q n , with the understanding that in each formula, either all u i denote p i or all denote q i .