On the Lorenz-Fibonacci sequences and substitutions

Abstract: In the present article, we consider two families of integer sequences, the $(k,r)$-Lorenz-Fibonacci sequences of the first and second kind, whose characteristic polynomial is $$p_{k,r}(x)=x^{k-x^{{k-1}-\cdots-}} x^{r+x^{r-1}+\cdots} +x+1,$$ where $k\geq 2r+1$, and $k\geq 3$. These families include the well-known $k$-bonacci sequence, when $r=0$. These sequences arise naturally in the study of the dynamics of the Lorenz attractor. We introduce two families of substitutions (in an alphabet of $k$ symbols) so that they are associated with each of the families of integer sequences and share the same polynomial. We study the main combinatorial properties of these substitutions.

• The paper constructs two families of Lorenz-Fibonacci sequences and substitutions governed by the characteristic polynomial pₖ,ᵣ(x), with generating functions, dominant-root asymptotics, and exact word-length correspondences.
• It shows that the substitution ζₖ,ᵣ has exactly r+1 periodic points of period r+1, while φₖ,ᵣ has a unique fixed point, despite both sharing the same incidence matrix and characteristic polynomial.
• The results demonstrate that identical spectral data can produce different combinatorial structures, while leaving connections to Lorenz-map dynamics, language complexity, and Binet-type formulas open for future work.

Overview

This paper by San Martín and Sirvent (2608.12739) studies two families of integer sequences — the $(k,r)$-Lorenz-Fibonacci sequences of first and second kind — together with two associated families of substitutions on a $k$-letter alphabet. Both families are governed by the characteristic polynomial

$$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$

where $k\geq 2r+1$ and $k\geq 3$. The polynomial arises from the symbolic dynamics of the geometric Lorenz attractor: prior work of the same authors showed that shift spaces satisfying certain combinatorial and dynamical assumptions are encoded by this family. The case $r=0$ recovers the classical $k$-bonacci sequence and substitution, so the paper can be read as a systematic generalization of $k$-bonacci combinatorics to the Lorenz setting.

A notable empirical observation is that most members of these families were not listed in the OEIS at the time of writing; only special cases such as $(3,0)$ (Tribonacci), $(4,0)$ (Tetranacci), $k$0, and $k$1 appear there.

The $k$2-Lorenz-Fibonacci sequences

The recurrence is

$k$3

with two choices of initial conditions. The first kind uses $k$4, $k$5; the second kind uses initial values that begin as powers of two, $k$6 for $k$7, followed by an explicit correction term for $k$8.

The paper establishes three structural results. First, Proposition 2 gives closed-form initial growth: for $k$9 one has exactly $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$0, and the negative terms of the recurrence only begin to affect values at $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$1, where $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$2. This quantifies precisely how long the sequence behaves like pure doubling before the "subtractive" part of the recurrence activates. Second, the generating function of the first-kind sequence has the compact form

$$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$3

and a parallel formula holds for the second kind. Third, since $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$4 was previously shown to have a single positive dominant root $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$5, both families grow asymptotically like $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$6. Binet-type formulas are explicitly not computed.

The substitutions $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$7 and their combinatorics

The first family of morphisms, $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$8 on $$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1,$$9, generalizes the $k\geq 2r+1$0-bonacci substitution ($k\geq 2r+1$1). For $k\geq 2r+1$2 the rules take the form $k\geq 2r+1$3 for odd $k\geq 2r+1$4, $k\geq 2r+1$5 for even $k\geq 2r+1$6, and $k\geq 2r+1$7 for $k\geq 2r+1$8, with $k\geq 2r+1$9. The main theorem (Theorem 1) establishes:

• Periodic structure: unlike the $k\geq 3$0-bonacci case, $k\geq 3$1 has no fixed points but exactly $k\geq 3$2 periodic points of period $k\geq 3$3, all lying on a single orbit generated by the odd symbols $k\geq 3$4.
• Word recurrence: the iterates satisfy a two-term decomposition mixing iterates of symbols $k\geq 3$5 and $k\geq 3$6, e.g. $k\geq 3$7 factors as a block of $k\geq 3$8 descending iterates $k\geq 3$9 followed by $r=0$0 iterates of symbol $r=0$1. Crucially, $r=0$2 cannot be written purely as a juxtaposition of words $r=0$3 — a qualitative departure from the $r=0$4-bonacci identity $r=0$5.
• Return time: the minimal return time of symbol $r=0$6 under $r=0$7 equals $r=0$8, matching the graph-theoretic minimum over cycles through vertex $r=0$9; any other substitution sharing the same graph has strictly larger return time. This minimality property is the stated reason for choosing this particular representative among substitutions with the same underlying graph.
• Spectral data: the characteristic polynomial of $k$0 is $k$1, proved via conjugation by the reversal permutation matrix to a companion-like matrix analyzed in the authors' earlier work.
• Sequence correspondence: $k$2, i.e., the word lengths reproduce the shifted first-kind sequence. The proof tracks the count $k$3 of occurrences of symbol $k$4, which stays zero until step $k$5 and then doubles — mirroring the closed-form initial behavior of $k$6.

The restriction $k$7 is essential here: the authors note that for $k$8 the substitution changes structure and several properties fail. The case $k$9, $k$0 is not even primitive.

The direct substitutions $k$1

The second family, $k$2, defined by $k$3 for $k$4, $k$5 for $k$6, and $k$7, shares the incidence matrix with $k$8 but exhibits different combinatorial behavior. For $k$9 it coincides with the $(3,0)$0-bonacci substitution; for $(3,0)$1 it does not.

Theorem 2 shows that $(3,0)$2 has a unique fixed point and no other periodic orbits — in sharp contrast to the $(3,0)$3 periodic points of $(3,0)$4. Its word recurrence again mixes iterates of $(3,0)$5 and $(3,0)$6, and again admits no pure juxtaposition decomposition into $(3,0)$7. Its characteristic polynomial is also $(3,0)$8, obtained via a Laplace expansion yielding the factorization $(3,0)$9 where $(4,0)$0. Finally, $(4,0)$1, linking this family to the second-kind sequences.

The structural distinction between the two families is made precise at the level of graphs: $(4,0)$2 and $(4,0)$3 are not isomorphic for $(4,0)$4, since vertex $(4,0)$5 has in-degree $(4,0)$6 in the former (with vertex $(4,0)$7 having in-degree $(4,0)$8) versus in-degree $(4,0)$9 in the latter. Thus two non-isomorphic substitutions carry identical incidence matrices yet generate sequences with the same characteristic polynomial but distinct initial conditions and fixed-point structure. The authors also note that $k$00 is obtained from the Lorenz shift graph of their earlier paper by reversing edge directions and relabelling vertices, confirming the dynamical provenance of these morphisms.

Limitations and open questions

The paper is candid about what lies outside its scope. The connection between the substitutions $k$01, $k$02 and the actual dynamics of Lorenz maps is asserted motivationally but not developed; the relationship between $k$03 and the growth of Markov partition elements for the corresponding Lorenz maps is likewise deferred. The languages generated by the fixed points — including their complexity functions — are not studied, although the authors indicate that $k$04 was selected among graph-sharing substitutions because its language has lower complexity. Binet formulas are omitted, and the free-group approach to describing word recurrences is mentioned as an unexplored alternative. Whether the complexity functions distinguish all substitutions sharing a given Lorenz-Fibonacci graph remains open.

Conclusion

The paper constructs two families of integer sequences and two structurally distinct families of substitutions, all governed by the single polynomial $k$05 arising from Lorenz symbolic dynamics. It computes generating functions, closed-form initial segments, return times, periodic-orbit counts, and exact correspondences between word lengths and sequence terms. The central finding is that equal spectral data do not force equal combinatorics: $k$06 and $k$07 share an incidence matrix yet differ in fixed points, periodic orbits, graphs, and admissible decompositions of iterates. The bridge from these combinatorial objects back to Lorenz dynamics remains the principal open problem.