Zeckendorf Representation Revisited

I've dealt with the topic of the Zeckendorf representation of numbers in earlier posts: namely The Fibonacci Number Base on July 30th 2019 and Goldbach's Conjecture and Zeckendorf's Theorem on November 15th 2015. As can be seen, the topic doesn't come up all that much and so, when it does, it's time to make a note of it because the opportunity may not come again for a long time. There are so many interesting topics in number theory that it's easy to forget even some of the really interesting ones like the Zeckendorf representation of a number.

The reason I was reminded was that yesterday I turned 27038 days old and one the properties of this number is that it's a member of OEIS A179250:

A179250

Numbers that have 10 terms in their Zeckendorf representation.

Such numbers are uncommon as the following list of the initial members attests:

10945, 15126, 16723, 17333, 17566, 17655, 17689, 17702, 17707, 17709, 17710, 21891, 23488, 24098, 24331, 24420, 24454, 24467, 24472, 24474, 24475, 26072, 26682, 26915, 27004, 27038, 27051, 27056, 27058, 27059, 27669, 27902, 27991

Notice how the numbers are clumped together. For example: 27004, 27038, 27051, 27056, 27058, 27059. This is seen more clearly in a plot. See Figure 1.

Figure 1: horizontal red lines added for emphasis

Before going on however, we should remind ourselves what the Zeckendorp representation is all about. Let's restate Zeckendorf's Theorem:

Every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.

The theorem has two parts:

Existence: every positive integer \(n\) has a Zeckendorf representation.

Uniqueness: no positive integer \(n\) has two different Zeckendorf representations.

Source 

So 27038 has ten terms in its Zeckendorf representation. What are these terms? We can use this site to quickly identify them. The terms are:

1, 3, 8, 21, 89, 233, 610, 1597, 6765, 17711 

and thus

27038 = 1 +3 + 8 + 21 + 89 + 233 + 610 + 1597 + 6765 + 17711 

The first few Fibonacci numbers are 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368 and so the Zeckendorf representation, using the same site as listed earlier, becomes:

\(27038 = 101001010101001010101_{Zeck} \text{ with 10 ones} \)

There are more such numbers coming up in the near future (27051, 27056, 27058 and 27059) but after that there is a big gap of 610 days to 27669. It can be noted that 27058 and 27059 are a consecutive pair. The thing about ten terms is that this number is relatively uncommon. Generally, fewer terms are required as can be seen from the following list of numbers from 27038 to 27045 (link) and this is in an area where such numbers are clumped together:

27038 = 101001010101001010101Zeck with 10 ones

27039 = 101001010101010000000Zeck with 7 ones

27035 = 101001010101001010001Zeck with 9 ones

27036 = 101001010101001010010Zeck with 9 ones

27037 = 101001010101001010100Zeck with 9 ones

27038 = 101001010101001010101Zeck with 10 ones

27039 = 101001010101010000000Zeck with 7 ones

27040 = 101001010101010000001Zeck with 8 ones

27041 = 101001010101010000010Zeck with 8 ones

27042 = 101001010101010000100Zeck with 8 ones

27043 = 101001010101010000101Zeck with 9 ones

27044 = 101001010101010001000Zeck with 8 ones

27045 = 101001010101010001001Zeck with 9 ones