THE AVERAGE GAP DISTRIBUTION FOR GENERALIZED ZECKENDORF DECOMPOSITIONS

被引:0
|
作者
Beckwith, Olivia [1 ]
Bower, Amanda [2 ]
Gaudet, Louis [3 ]
Insoft, Rachel [4 ]
Li, Shiyu [5 ]
Miller, Steven J. [6 ]
Tosteson, Philip [6 ]
机构
[1] Harvey Mudd Coll, Dept Math, Claremont, CA 91711 USA
[2] Univ Michigan, Dept Math & Stat, Dearborn, MI 48128 USA
[3] Yale Univ, Dept Math, New Haven, CT 06510 USA
[4] Wellesley Coll, Dept Math, Wellesley, MA 02481 USA
[5] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA
[6] Williams Coll, Dept Math & Stat, Williamstown, MA 01267 USA
来源
FIBONACCI QUARTERLY | 2013年 / 51卷 / 01期
基金
美国国家科学基金会;
关键词
D O I
暂无
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
An interesting characterization of the Fibonacci numbers is that if we write them as F-1 = 1, F-2 = 2, F-3 = 3, F-4 = 5,..., then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is now known as Zeckendorf's Theorem [21], and similar decompositions exist for many other sequences {G(n+1) = c(1) G(n) + . . . + c(L)G(n+1-L)} arising from recurrence relations. Much more is known. Using continued fraction approaches, Lekkerkerker [15] proved the average number of summands needed for integers in [G(n), G(n+1)) is on the order of C(Lek)n for a non-zero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean. Kologlu, Kopp, Miller and Wang [13, 17] recently recast the problem combinatorially, reproving and generalizing these results. We use this new perspective to investigate the distribution of gaps between summands. We explore the average behavior over all m is an element of [G(n), G(n+1)) for special choices of the c(i)'s. Specifically, we study the case where each c(i) is an element of{0, 1} and there is a g such that there are always exactly g - 1 zeros between two non-zero c(i)'s; note this includes the Fibonacci, Tribonacci and many other important special cases. We prove there are no gaps of length less than g, and the probability of a gap of length j > g decays geometrically, with the decay ratio equal to the largest root of the recurrence relation. These methods are combinatorial and apply to related problems; we end with a discussion of similar results for far-difference (i.e., signed) decompositions.
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页码:13 / 27
页数:15
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