## Involve: A Journal of Mathematics

• Involve
• Volume 10, Number 1 (2017), 125-150.

### A generalization of Zeckendorf's theorem via circumscribed $m$-gons

#### Abstract

Zeckendorf’s theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy $F1 = 1$, $F2 = 2$, and $Fn = Fn−1 + Fn−2$ for $n ≥ 3$. The distribution of the number of summands in such a decomposition converges to a Gaussian, the gaps between summands converge to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work is needed to assume the coefficients in the recurrence relation are nonnegative and the first term is positive.

We extend these results by creating an infinite family of integer sequences called the $m$-gonal sequences arising from a geometric construction using circumscribed $m$-gons. They satisfy a recurrence where the first $m+1$ leading terms vanish, and thus cannot be handled by existing techniques. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the decompositions and prove that it displays Gaussian behavior. There is geometric decay in the distribution of gaps, both for gaps taken from all integers in an interval and almost surely in distribution for the individual gap measures associated to each integer in the interval. We end by proving that the distribution of the longest gap between summands is strongly concentrated about its mean, behaving similarly as in the longest run of heads in tosses of a coin.

#### Article information

Source
Involve, Volume 10, Number 1 (2017), 125-150.

Dates
Revised: 5 December 2015
Accepted: 13 December 2015
First available in Project Euclid: 22 November 2017

https://projecteuclid.org/euclid.involve/1511371087

Digital Object Identifier
doi:10.2140/involve.2017.10.125

Mathematical Reviews number (MathSciNet)
MR3561734

Zentralblatt MATH identifier
1348.11015

#### Citation

Dorward, Robert; Ford, Pari L.; Fourakis, Eva; Harris, Pamela E.; Miller, Steven J.; Palsson, Eyvindur; Paugh, Hannah. A generalization of Zeckendorf's theorem via circumscribed $m$-gons. Involve 10 (2017), no. 1, 125--150. doi:10.2140/involve.2017.10.125. https://projecteuclid.org/euclid.involve/1511371087

#### References

• H. Alpert, “Differences of multiple Fibonacci numbers”, Integers 9:6 (2009), 745–749.
• O. Beckwith, A. Bower, L. Gaudet, R. Insoft, S. Li, S. J. Miller, and P. Tosteson, “The average gap distribution for generalized Zeckendorf decompositions”, Fibonacci Quart. 51:1 (2013), 13–27.
• I. Ben-Ari and S. J. Miller, “A probabilistic approach to generalized Zeckendorf decompositions”, preprint, 2014.
• A. Best, P. Dynes, X. Edelsbrunner, B. McDonald, S. J. Miller, K. Tor, C. Turnage-Butterbaugh, and M. Weinstein, “Gaussian distribution of the number of summands in generalized Zeckendorf decomposition in small intervals”, Integers 16 (2016), Paper A6.
• A. Bower, R. Insoft, S. Li, S. J. Miller, and P. Tosteson, “The distribution of gaps between summands in generalized Zeckendorf decompositions”, J. Combin. Theory Ser. A 135 (2015), 130–160.
• M. Catral, P. L. Ford, P. E. Harris, S. J. Miller, and D. Nelson, “Generalizing Zeckendorf's theorem: the Kentucky sequence”, Fibonacci Quart. 52:5 (2014), 69–91.
• M. Catral, P. L. Ford, P. E. Harris, S. J. Miller, and D. Nelson, “Legal decompositions arising from non-positive linear recurrences”, preprint, 2016.
• M. Catral, P. L. Ford, P. E. Harris, S. J. Miller, D. Nelson, Z. Pan, and H. Xu, “New behavior in legal decompositions arising from non-positive linear recurrences”, preprint, 2016.
• D. E. Daykin, “Representation of natural numbers as sums of generalised Fibonacci numbers”, J. London Math. Soc. 35 (1960), 143–160.
• P. Demontigny, T. Do, A. Kulkarni, S. J. Miller, D. Moon, and U. Varma, “Generalizing Zeckendorf's theorem to $f$-decompositions”, J. Number Theory 141 (2014), 136–158.
• P. Demontigny, T. Do, A. Kulkarni, S. J. Miller, and U. Varma, “A generalization of Fibonacci far-difference representations and Gaussian behavior”, Fibonacci Quart. 52:3 (2014), 247–273.
• R. Dorward, P. L. Ford, E. Fourakis, P. E. Harris, S. J. Miller, E. Palsson, and H. Paugh, “Individual gap measures from generalized Zeckendorf decompositions”, preprint, 2015.
• M. Drmota and J. Gajdosik, “The distribution of the sum-of-digits function”, J. Théor. Nombres Bordeaux 10:1 (1998), 17–32.
• P. Filipponi, P. J. Grabner, I. Nemes, A. Pethö, and R. F. Tichy, “Corrigendum to: “Generalized Zeckendorf expansions””, Appl. Math. Lett. 7:6 (1994), 25–26.
• P. J. Grabner and R. F. Tichy, “Contributions to digit expansions with respect to linear recurrences”, J. Number Theory 36:2 (1990), 160–169.
• P. J. Grabner, R. F. Tichy, I. Nemes, and A. Pethö, “Generalized Zeckendorf expansions”, Appl. Math. Lett. 7:2 (1994), 25–28.
• T. J. Keller, “Generalizations of Zeckendorf's theorem”, Fibonacci Quart. 10:1 (1972), 95–102, 111, 112.
• M. Koloğlu, G. S. Kopp, S. J. Miller, and Y. Wang, “On the number of summands in Zeckendorf decompositions”, Fibonacci Quart. 49:2 (2011), 116–130.
• T. Lengyel, “A counting based proof of the generalized Zeckendorf's theorem”, Fibonacci Quart. 44:4 (2006), 324–325.
• S. J. Miller and Y. Wang, “From Fibonacci numbers to central limit type theorems”, J. Combin. Theory Ser. A 119:7 (2012), 1398–1413.
• S. J. Miller and Y. Wang, “Gaussian behavior in generalized Zeckendorf decompositions”, pp. 159–173 in Combinatorial and additive number theory: CANT 2011 and 2012 (New York, NY, 2011–2012), edited by M. B. Nathanson, Springer Proceedings in Mathematics and Statistics 101, Springer, New York, NY, 2014.
• M. F. Schilling, “The longest run of heads”, College Math. J. 21:3 (1990), 196–207.
• W. Steiner, “Parry expansions of polynomial sequences”, Integers 2 (2002), Paper A14.
• W. Steiner, “The joint distribution of greedy and lazy Fibonacci expansions”, Fibonacci Quart. 43:1 (2005), 60–69.
• E. Zeckendorf, “Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas”, Bull. Soc. Roy. Sci. Liège 41 (1972), 179–182.