@MISC{Strazdins98fibonacciand, author = {Indulis Strazdins}, title = {FIBONACCI AND LUCAS NUMBERS}, year = {1998} }

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Abstract

We present here two sieve-type explicit formulas for r-Fibonacci and r-Lucas numbers (r = 2,3,...) that connect them with families of well-defined combinatorial numbers, and discuss some particular cases. 1. DEFINITIONS We consider the two main families of sequences {F „ (r) } and {L^} (r = 2,3,...), determined by the simplest general r*-order linear recursion (<2 ^ denotes either F ^ or 1$) with initial conditions & r) = E G & («*/•) (i) F ^ = 0, FW = 1,..., F/> = 2'- 2 (2 < j < r-1); (2) W=r,W = \,...,Lf = 2J-l (l<j<r-l). (3) F ^ and 1 $ are r-Fibonacci and r-Lucas numbers, respectively (cf. [2], [6], [8], [9]; also [7] with ai-1 for all i)—or the "fundamental " and "primordial " sequences named by Lucas. The sequences {F^} and {L^} differ from the known Tribonacci, Tetranacci, etc., sequences in having a shift r- 2 places backwards. The recursion (1) implies a fundamental property—the subtraction law Q [ r) = 2QW1-QW_i {n>r + 2) (4) for sequences of both kinds. Our aim is to evaluate the differences 2 n ~ 2- Fw (r) and 2 "- 1- 1 $ caused by this subtraction. We propose a method of exact calculation of Fw (r) and Z ^ r). As a result, explicit formulas (12) and (18) are obtained, which generalize the known formulas in the particular case r = 2 (Section 4). The evaluation of 2 n ~ 2 2 * THE r-FIBONACCI SEQUENCES- F ^ r) involves a family of numbers d(m,l) = l, d(m,n) = n i ^ n_2 J2 (5) _ 2fn + n-3(in+n-3\0„-2 m-l { m m-2 3) 2 " ' 2