Factorizations of the Fibonacci Infinite Word

Gabriele Fici Dipartimento di Matematica e Informatica Università di Palermo Via Archirafi 34 90123 Palermo Italy

                    Gabriele.Fici@unipa.it

Abstract The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers.

1 Preliminaries

The well-known sequence of Fibonacci numbers (sequence \underline{A000045} in the On-Line Encyclopedia of Integer Sequences) is defined by F_{1} = 1, F_{2} = 1 and for every n > 2, F_{n} = F_{n−1} + F_{n−2}. The first few values of the sequence F_{n} are reported in Table 1 for reference.

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ F_{n} & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 & 144 & 233 & 377 & 610 & 987 & 1597 & 2584 & 4181 & 6765 \end{matrix}

Table 1: The first few values of the sequence of Fibonacci numbers.

A basic property of Fibonacci numbers (that can be easily proved by induction) is that 1 plus the sum of the first n Fibonacci numbers is equal to the (n + 2)-th Fibonacci number:

1 + ∑_{i=1}^{n} F_{i} = F_{n+2}.  (1)

We recall here a famous result, usually attributed to Zeckendorf [13], but published earlier by Lekkerkerker [8] and which, in fact, is a special case of an older and more general result due to Ostrowski [10]. It permits us to use Fibonacci numbers as a basis for representing integers:

Theorem 1. Every positive integer can be expressed uniquely as the sum of one or more distinct non-consecutive Fibonacci numbers F_{n}, n > 1.

For example, 17 = 13 + 3 + 1 = F_{7} + F_{4} + F_{2}, and there is no other way to write 17 as the sum of non-consecutive Fibonacci numbers (assuming the convention that F_{1} is not used in the representation). Thus, one can represent natural numbers as strings of 0-1 bits, where the i-th bit (from the right) encodes the presence/absence of the (i+1)-th Fibonacci number in the representation given by Theorem 1. So for example the number 17 is represented by 100101. We call this representation of natural numbers the Zeckendorf representation.

The first few natural numbers and their Zeckendorf representations are displayed in Table 2, where we padded to the left with 0s in order to have strings of the same length. Note that with 6 bits one can represent the first 21 natural numbers. In fact, for every n > 0, there are exactly F_{n} integers whose leftmost 1 in the Zeckendorf representation is in position n (starting from the right). From (1), we derive that one needs n bits to represent the first F_{n+2} natural numbers.

The strings of length n forming the Zeckendorf representations of the first F_{n+2} natural numbers are precisely all the 0-1 strings of length n not containing two consecutive 1s. These strings are in lexicographic order if the natural numbers are in increasing order from 0 to F_{n+2} − 1.

Zeck. decimal Zeck. decimal Zeck. decimal
000000 0  010000 8  100100 16
000001 1  010001 9  100101 17
000010 2  010010 10  101000 18
000100 3  010100 11  101001 19
000101 4  010101 12  101010 20
001000 5  100000 13
001001 6  100001 14
001010 7  100010 15

Table 2: The Zeckendorf representations of the first few natural numbers coded with 6 bits.

Let us define f(n), for every n ≥ 0, as the rightmost digit of the Zeckendorf representation of n. For every n > 1 we define the n-th Fibonacci word as the string

f_{n} = f(0)f(1) ⋯ f(F_{n} − 1)

of length |f_{n}| = F_{n}. By convention, we set f_{1} = 1. The first few Fibonacci words are shown in Table 3.

\begin{aligned}f_{1} & = 1 \\ f_{2} & = 0 \\ f_{3} & = 01 \\ f_{4} & = 010 \\ f_{5} & = 01001 \\ f_{6} & = 01001010 \\ f_{7} & = 0100101001001 \\ f_{8} & = 010010100100101001010 \\ f_{9} & = 0100101001001010010100100101001001 \end{aligned}

Table 3: The first few Fibonacci words.

We also define the Fibonacci infinite word f as the limit of f_{n} as n goes to infinity. That is, f is the infinite word whose n-th letter is the “parity” of the Zeckendorf representation of n:

f = f(0)f(1)f(2)f(3) ⋯ = 0100101001001010010 ⋯

In the Zeckendorf representation of an integer, when the n-th digit from the right is a 1, the (n − 1)-th digit from the right is a 0. Hence, the rightmost n − 2 digits of the Zeckendorf representations of the natural numbers from F_{n+1} to F_{n+2} − 1 are the same rightmost n − 2 digits of the Zeckendorf representations of the first F_{n} natural numbers. For example, the 2 rightmost digits of the Zeckendorf representations of 5, 6 and 7 are, respectively, 00, 01, 10, as well as the 2 rightmost digits of the Zeckendorf representations of the 0, 1 and 2. We deduce that for every n > 2, one has

f_{n} = f_{n−1}f_{n−2}.  (2)

For more details on Fibonacci words the reader can see, for instance, [1].

Recall that a factorization of an infinite word w is a sequence (x_{n})_{n≥1} of finite words such that w can be expressed as the concatenation of the elements of the sequence, i.e., w = ∏_{n≥1} x_{n}.

In general, exhibiting a factorization (x_{n})_{n≥1} of an infinite word w can be useful to better understand the combinatorics of w, provided the sequence (x_{n})_{n≥1} has non-trivial combinatorial properties—for example, all the words in the sequence are palindromes, squares, or prefixes of w.

Another point of view consists in defining a factorization by some general rule that can be applied to any infinite word. The sequence (x_{n})_{n≥1} is therefore determined by the particular instance of the infinite word w (as is the case, for example, in the Lempel-Ziv factorization or in the Lyndon factorization, that we will see below). In this case, the word w can have particular properties that make it a limit example for that particular factorization.

In next sections, we will show a number of factorizations of the Fibonacci infinite word that make use of the Fibonacci finite words and other related words. These factorizations have been introduced over the time in different papers, and we think it can be useful to collect them all together for reference. We also add some (at least to the best of our knowledge) novel factorizations. Moreover, we present these factorizations in an order that allows us to provide a short and elementary proof for each of them, despite the original proofs being sometimes more involved or more technical.

2 Fibonacci words and co-Fibonacci words

The first factorization of the Fibonacci infinite word we exhibit is the following.

Proposition 2. The Fibonacci infinite word can be obtained by concatenating 0 and the Fibonacci words:

f = 0 ∏_{n≥1} f_{n}  (3)
= 0 · 1 · 0 · 01 · 010 · 01001 · 01001010 ⋯

Proof. Since for every i ≥ 1, |f_{i}| = F_{i}, it is sufficient to prove that, for every n ≥ 1, f_{n} occurs in f starting at position 1 + ∑_{i=1}^{n−1} F_{i} = F_{n+1}. From (2), we have f_{n+2} = f_{n+1}f_{n}, so that f_{n} has an occurrence in f starting at position |f_{n+1}| = F_{n+1}.

Let us consider the sequence p_{n} of the palindromic prefixes of f, also called central words. The first few values of the sequence p_{n} are displayed in Table 4, where ε denotes the empty word, i.e., the word of length 0.

\begin{aligned}p_{3} & = ε \\ p_{4} & = 0 \\ p_{5} & = 010 \\ p_{6} & = 010010 \\ p_{7} & = 01001010010 \\ p_{8} & = 0100101001001010010 \\ p_{9} & = 01001010010010100101001001010010 \end{aligned}

Table 4: The first few central words.

As it is well-known, for every n ≥ 3, p_{n} is obtained from f_{n} by removing the last two letters. More precisely, we have for every n ≥ 1,

f_{2n+1} = p_{2n+1}01, f_{2n+2} = p_{2n+2}10.  (4)

The fundamental property of the central words is the following:

Lemma 3. For every n ≥ 2 one has

p_{2n+1} = p_{2n−1}01p_{2n} = p_{2n}10p_{2n+1}, p_{2n+2} = p_{2n}10p_{2n+1} = p_{2n+1}01p_{2n}.

Proof. Follows immediately from (2) and (4).

Remark 4. It is easy to see from (2) that for every n ≥ 4, one has f_{n} = f_{n−2}f_{n−3}f_{n−2}. We have therefore from (3):

f = 01001 ∏_{n≥2} f_{n}f_{n−1}f_{n}  (5)
= 01001 · (0 · 1 · 0)(01 · 0 · 01)(010 · 01 · 010)(01001 · 010 · 01001) ⋯

Analogously, since 1 = f_{3}, we can write

f = 0100 ∏_{n≥2} f_{n−1}f_{n}f_{n−1}  (6)
= 0100 · (1 · 0 · 1)(0 · 01 · 0)(01 · 010 · 01)(010 · 01001 · 010) ⋯

We now introduce a class of words that we call the co-Fibonacci words. Although this class has appeared previously in the literature [2], to the best of our knowledge no one has yet given a name to them.

The co-Fibonacci words f′_{n} are defined by complementing the last two letters in the Fibonacci words f_{n}, that is, f′_{n} = p_{n}yx, where x and y are the letters such that f_{n} = p_{n}xy. Equivalently, co-Fibonacci words can be defined by f′_{n} = f_{n−2}f_{n−1} for every n ≥ 3. The first few co-Fibonacci words are displayed in Table 5.

The following lemma is a direct consequence of Lemma 3.

Lemma 5. For every n ≥ 2 one has

f′_{2n+1} = f_{2n}f′_{2n−1}, f_{2n+2} = f_{2n}f′_{2n+1}.

Proposition 6. The Fibonacci word can be obtained by concatenating 0 and the odd co-Fibonacci words:

f = 0 ∏_{n≥1} f′_{2n+1}  (7)
= 0 · 10 · 01010 · 0100101001010 ⋯

\begin{aligned}f′_{3} & = 10 \\ f′_{4} & = 001 \\ f′_{5} & = 01010 \\ f′_{6} & = 01001001 \\ f′_{7} & = 0100101001010 \\ f′_{8} & = 010010100100101001010 \end{aligned}

Table 5: The first few co-Fibonacci words.

Proof. Follows directly from (3) replacing f_{2n−1}f_{2n} with f′_{2n+1}.

Analogously, we have the following:

Proposition 7. The Fibonacci word can be obtained by concatenating 01 and the even co-Fibonacci words:

f = 01 ∏_{n≥1} f′_{2n+2}  (8)
= 01 · 001 · 01001001 · 010010100100101001010 ⋯

Proof. Follows directly from (3) replacing f_{2n}f_{2n+1} with f′_{2n+2}.

3 Singular words

Let us define the left rotation of a non-empty word w = w_{1}w_{2} ⋯ w_{n}, w_{i} letters, as the word w^{λ} = w_{n}w_{1} ⋯ w_{n−1}. Analogously, the right rotation of w is defined as the word w^{ρ} = w_{2} ⋯ w_{n}w_{1}.

The singular words f̂_{n} are defined by complementing the first letter in the left rotations of the Fibonacci words f_{n}. The first few singular words are displayed in Table 6. Note that for every n ≥ 1, one has f̂_{2n+1} = 0p_{2n+1}0 and f̂_{2n+2} = 1p_{2n+2}1 .

The singular words are palindromic factors of f but do not appear as prefixes of f (by the way, f also contains other palindromic factors besides the central words p_{n} and the singular words f̂_{n}, e.g., 1001 or 01010, see [6] for more details). Their name comes from the fact that among the F_{n} + 1 factors of f of length F_{n}, there are F_{n} of them that can be obtained one from each other by iteratively applying left (or equivalently right) rotation and one, the singular word, whose left (or equivalently right) rotation is not a factor of f.

Wen and Wen [12] proved that the Fibonacci infinite word can be obtained by concatenating the singular words:

\begin{aligned}f̂_{1} & = 0 \\ f̂_{2} & = 1 \\ f̂_{3} & = 00 \\ f̂_{4} & = 101 \\ f̂_{5} & = 00100 \\ f̂_{6} & = 10100101 \end{aligned}

Table 6: The first few singular words.

Proposition 8. The Fibonacci infinite word is the concatenation of the singular words:

f = ∏_{n≥1} f̂_{n}  (9)
= 0 · 1 · 00 · 101 · 00100 · 10100101 ⋯

Proof. Indeed, (9) follows directly from (3) and the definition of singular words, observing that the Fibonacci words end by letter 0 and 1 alternatingly.

The factorization (9) is in fact the Lempel-Ziv factorization of f. The Lempel-Ziv factorization is a factorization widely used in computer science for compressing strings [14]. The Lempel-Ziv factorization of a word w is w = w_{1}w_{2} ⋯ where w_{1} is the first letter of w and for every i ≥ 2, w_{i} is the shortest prefix of w_{i}w_{i+1} ⋯ that occurs only once in the word w_{1}w_{2} ⋯ w_{i}. Roughly speaking, at each step one searches for the shortest factor that did not appear before. Remark 9. It is easy to see, using for example (2), (4) and the definition of singular words, that for every n ≥ 4, f̂_{n} = f̂_{n−2}f̂_{n−3}f̂_{n−2}. Therefore, from (9), we have

f = 0100 ∏_{n≥2} f̂_{n}f̂_{n−1}f̂_{n}  (10)
= 0100 · (1 · 0 · 1)(00 · 1 · 00)(101 · 00 · 101)(00100 · 101 · 00100) ⋯

Since 0 = f̂_{1}, we hence obtain

f = 010 ∏_{n≥2} f̂_{n−1}f̂_{n}f̂_{n−1}  (11)
= 010 · (0 · 1 · 0)(1 · 00 · 1)(00 · 101 · 00)(101 · 00100 · 101) ⋯

The factorization (11) is a sort of dual with Lucas numbers of the factorization in singular words (9). Indeed, the sequence of factor lengths in (9) is the sequence of Fibonacci numbers, while if in (11) one decomposes the first term as 01 · 0, then the sequence of factor lengths is the sequence of Lucas numbers \underline{A000032}: 2,1,3,4,7,11, etc.

Figure 1: The lower Christoffel word c_{7} = 0010010100101 (left) and the upper Christoffel word c̃_{7} = 1010010100100 (right) are the best grid approximations, respectively from above and from below, of the Euclidean segment joining the points (0, 0) and (8, 5) = (F_{6}, F_{5}).

4 Christoffel words

The lower Christoffel words are defined by c_{n} = 0p_{n}1, for every n ≥ 3. The lower Christoffel words are the Lyndon factors of f, i.e., they are lexicographically smaller than any of their proper suffixes (with respect to the order induced by 0 < 1).

\begin{aligned}c_{3} & = 01 \\ c_{4} & = 001 \\ c_{5} & = 00101 \\ c_{6} & = 00100101 \\ c_{7} & = 0010010100101 \\ c_{8} & = 001001010010010100101 \end{aligned}

Table 7: The first few lower Christoffel words.

If in the Euclidean plane one interprets each 0 by a horizontal unitary step and each 1 with a vertical unitary step, the lower Christoffel word c_{n} is the best grid approximation from below of the segment joining the point (0, 0) to the point (F_{n−1}, F_{n−2}) (see Figure 1).

Analogously, the upper Christoffel words are defined by c̃_{n} = 1p_{n}0, for every n ≥ 3. Therefore, the upper Christoffel words are the reversals of the lower Christoffel words (we use the notation w̃ for the reversal, a.k.a. mirror image, of the word w). The upper Christoffel words are the anti-Lyndon factors of f, i.e., they are lexicographically greater than any of their proper suffixes (with respect to the order induced by 0 < 1). Moreover, c̃_{n} is the best grid approximation from above of the segment joining the point (0, 0) to the point (F_{n−1}, F_{n−2}).

Remark 10. For every n ≥ 3, c_{n} = f_{n}^{λ} if n is even, c_{n} = f̃_{n}^{λ} if n is odd. Therefore, c̃_{n} = f̃_{n}^{λ} if n is even, c̃_{n} = f_{n}^{λ} if n is odd.

\begin{aligned}c̃_{3} & = 10 \\ c̃_{4} & = 100 \\ c̃_{5} & = 10100 \\ c̃_{6} & = 10100100 \\ c̃_{7} & = 1010010100100 \\ c̃_{8} & = 101001010010010100100 \end{aligned}

Table 8: The first few upper Christoffel words.

Lemma 11. For every n ≥ 2 one has

c_{2n+1} = c_{2n}c_{2n−1}, c_{2n+2} = c_{2n}c_{2n+1},

and therefore

{ \atop c_{2n+1}} = { \atop c_{2n−1}}c̃_{2n}, { \atop c_{2n+2}} = { \atop c_{2n+1}}c̃_{2n}.

Proof. The first part follows from Lemma 5 by applying the right rotation to each side of the equalities. The second part follows from the first by applying the reversal.

The following result states that every Christoffel word is the product of two singular words. Lemma 12. For every n ≥ 1 one has

• c_{2n+1} = f̂_{2n−1}f̂_{2n}

• c_{2n+2} = f̂_{2n+1}f̂_{2n} and therefore

• { \atop c_{2n+1}} = f̂_{2n}f̂_{2n−1}

• { \atop c_{2n+2}} = f̂_{2n}f̂_{2n+1}

Proof. Follows directly from Lemma 3 and the definitions of Christoffel and singular words.

Melançon [9] proved that the Fibonacci word is the concatenation of the odd lower Christoffel words:

Proposition 13. The Fibonacci word is the concatenation of the odd lower Christoffel words:

f = ∏_{n≥1} c_{2n+1}  (12)
= 01 · 00101 · 0010010100101 ⋯

Proof. Follows directly from (9) and Lemma 12.

Actually, Melançon proved that (12) is precisely the Lyndon factorization of f. Recall that the Lyndon factorization of a word w is w = ℓ_{1}ℓ_{2} ⋯, where each ℓ_{i} is a Lyndon word and is lexicographically greater than or equal to ℓ_{i+1}. The uniqueness of such a factorization for finite words is a well-known theorem of Chen, Fox and Lyndon [4]. Siromoney et al. extended this factorization to infinite words [11].

Symmetrically, we have the following:

Proposition 14. The Fibonacci word is the concatenation of 0 and the even upper Christoffel words:

f = 0 ∏_{n≥2} c̃_{2n}  (13)
= 0 · 100 · 10100100 · 101001010010010100100 ⋯

Proof. Follows directly from (9) and Lemma 12.

In fact, it is easy to see that (13) is the Lyndon factorization of f if one takes the order induced by 1 < 0.

We now present two other factorizations based on Christoffel words. To the best of our knowledge, these factorizations did not appear before in literature.

Proposition 15. The Fibonacci word is the concatenation of 010 and the lower Christoffel words where each odd lower Christoffel word is squared:

f = 010 ∏_{n≥1} c_{2n+1}^{2}c_{2n+2}  (14)
= 010 · (01 · 01 · 001)(00101 · 00101 · 00100101) ⋯

Proof. Follows directly from (6) and Lemma 12. Indeed, by Lemma 12, we have

c_{2n+1}c_{2n+1}c_{2n+2} = f̂_{2n−1}f̂_{2n} · f̂_{2n−1}f̂_{2n} · f̂_{2n+1}f̂_{2n} = (f̂_{2n−1}f̂_{2n}f̂_{2n−1})(f̂_{2n}f̂_{2n+1}f̂_{2n}).

Analogously, we have the following:

Proposition 16. The Fibonacci word is the concatenation of 0100 and the upper Christoffel words where each even upper Christoffel word is squared:

f = 0100 ∏_{n≥1} { \atop c_{2n+1}}{ \atop c_{2n+2}}^{2}  (15)
= 0100 · (10 · 100 · 100)(10100 · 10100100 · 10100100) ⋯

Proof. Follows directly from (5) and Lemma 12. Indeed, by Lemma 12, we have

{ \atop c_{2n+1}}{ \atop c_{2n+2}}{ \atop c_{2n+2}} = f̂_{2n}f̂_{2n−1} · f̂_{2n}f̂_{2n+1} · f̂_{2n}f̂_{2n+1} = (f̂_{2n}f̂_{2n−1}f̂_{2n})(f̂_{2n+1}f̂_{2n}f̂_{2n+1}).

5 Reversals of Fibonacci words

One of the most known factorizations of the Fibonacci infinite word, and perhaps the most surprising, is the following.

Proposition 17. The Fibonacci word can be obtained also by concatenating the reversals of the Fibonacci words:

f = ∏_{n≥2} f̃_{n}  (16)
= 0 · 10 · 010 · 10010 · 01010010 ⋯

Proof. It follows from the definitions that taking the right rotation of f̃_{n} and complementing the last letter produces the n-th singular word f̂_{n}. Therefore, (16) follows directly from (9) observing that the reversals of the Fibonacci words start with 0 and 1 alternatingly.

The factorization (16) is basically the Crochemore factorization of f—the only difference is that the Crochemore factorization starts with 0, 1, 0 and then coincides with the one above (see [3]). Recall that the Crochemore factorization of w is w = c_{1}c_{2} ⋯ where c_{1} is the first letter of w and for every i > 1, c_{i} is either a fresh letter or the longest prefix of c_{i}c_{i+1} ⋯ occurring twice in f_{1}f_{2} ⋯ f_{i}. For example, the Crochemore factorization of the word w = 0101001 is 0 · 1 · 010 · 01, since 010 occurs twice in 01010.

In 1995, de Luca [5] considered the following factorization:

Proposition 18. The Fibonacci word can be obtained by concatenating the reversals of the even Fibonacci words.

f = ∏_{n≥2} f̃_{2n}  (17)
= 010 · 01010010 · 010100101001001010010 ⋯

Proof. Applying the reversal to (2), we have that f̃_{n} = f̃_{n−2}f̃_{n−1}, for every n > 2. So (17) follows directly from (16) by replacing { \atop f_{2n−2}}{ \atop f_{2n−1}} with f̃_{2n}.

In [5] de Luca proved that the factorization (17) has the following minimal property with respect to the lexicographical order: any non-trivial permutation of a finite number of the factors will produce an infinite word that is lexicographically greater than f.

Concatenating the reversals of the odd Fibonacci words instead of even ones still produces the Fibonacci word, if one prepends a 0:

Proposition 19. The Fibonacci word can be obtained by concatenating 0 and the reversals of the odd Fibonacci words:

f = 0 ∏_{n≥2} { \atop f_{2n+1}}  (18)
= 0 · 10010 · 1001001010010 ⋯

Proof. Follows directly from (16) by replacing { \atop f_{2n−1}}f̃_{2n} with { \atop f_{2n+1}}.

Recently [7], studying the so-called open and closed words, the following factorization has been proved:

Proposition 20. The Fibonacci word can be obtained by concatenating 01 and the squares of the reversals of the Fibonacci words:

f = 01 ∏_{n≥2}(f̃_{n})^{2}  (19)
= 01 · (0 · 0)(10 · 10)(010 · 010)(10010 · 10010) ⋯

Proof. Recalling that for every n ≥ 3, one has f̃_{n} = f̃_{n−2}f̃_{n−1}, we have, from (16), that f = f̃_{2}f̃_{3}f̃_{4} ⋯ = 0 · f̃_{1}f̃_{2}f̃_{2}f̃_{3} ⋯ = 01 · (f̃_{2}f̃_{2})(f̃_{3}f̃_{3}) ⋯ = 01 ∏_{n≥2}(f̃_{n})^{2}.

6 Generalization to standard Sturmian words

The Fibonacci word is the most prominent example of a standard Sturmian word. Let α be an irrational number such that 0 < α < 1, and let [0; d_{1} + 1, d_{2}, d_{3}, …] be the continued fraction expansion of α. The sequence of words defined by s_{1} = 1, s_{2} = 0 and s_{n} = s_{n−1}^{d_{n−2}}s_{n−2} for n ≥ 3, converges to the infinite word w_{α}, called the standard Sturmian word of slope α. The sequence of words s_{n} is called the standard sequence of w_{α}. The Fibonacci word is the standard Sturmian word of slope α = (3 − \sqrt{5})/2 and its standard sequence is the sequence of Fibonacci finite words, since one has d_{i} = 1 for every i ≥ 1.

Most of the factorizations we described in this note can be generalized to any standard Sturmian word. However, the proofs become more technical and less easy to present.

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2010 Mathematics Subject Classification: Primary 68R15. Keywords: Fibonacci word; Zeckendorf representation; Lyndon factorization; Lempel-Ziv factorization; Crochemore factorization.

(Concerned with sequences \underline{A000032} and \underline{A000045}.)

Received July 30 2015; revised versions received August 3 2015; August 6 2015. Published in Journal of Integer Sequences, August 7 2015.

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