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Recursions, Trains, Trees, and Combinatorial Rod Set Algebra

Ethan D. Bolker, Debra K. Borkovitz, Katelyn Lee

TL;DR

This work develops a combinatorial rod-set framework to study linear recursions. By encoding recurrences as expansions of rod sets through trees, and organizing these via an equivalence relation that cancels rod/antirod pairs, the authors define a formal rod-set algebra linked to generating functions. The approach yields unified proofs and new insights into classic sequences (Fibonacci, Lucas, Padovan), divisibility properties, and identities, and extends to analyzing Borwein trinomials and periodicity through rational generating functions and cyclotomic factors. Duality and scaling further connect identities with compositions and provide a versatile toolkit for exploring expansions to shapes and their arithmetic consequences.

Abstract

We explore a physical model of ordered sums of integers as trains of rods. The trains for a fixed, possibly infinite, set of rod lengths naturally correspond to nodes in a tree; relations among finite linear recursions encoded in the subtrees define algebraic operations on sets of rods. We use this algebra to prove classic identities for recursively defined sequences, to show that Lucas sequences are divisibility sequences, to characterize two-term linear Fibonacci identities, and to find the cyclotomic polynomial factors of Borwein trinomials. We complement abstractions with lots of examples.

Recursions, Trains, Trees, and Combinatorial Rod Set Algebra

TL;DR

This work develops a combinatorial rod-set framework to study linear recursions. By encoding recurrences as expansions of rod sets through trees, and organizing these via an equivalence relation that cancels rod/antirod pairs, the authors define a formal rod-set algebra linked to generating functions. The approach yields unified proofs and new insights into classic sequences (Fibonacci, Lucas, Padovan), divisibility properties, and identities, and extends to analyzing Borwein trinomials and periodicity through rational generating functions and cyclotomic factors. Duality and scaling further connect identities with compositions and provide a versatile toolkit for exploring expansions to shapes and their arithmetic consequences.

Abstract

We explore a physical model of ordered sums of integers as trains of rods. The trains for a fixed, possibly infinite, set of rod lengths naturally correspond to nodes in a tree; relations among finite linear recursions encoded in the subtrees define algebraic operations on sets of rods. We use this algebra to prove classic identities for recursively defined sequences, to show that Lucas sequences are divisibility sequences, to characterize two-term linear Fibonacci identities, and to find the cyclotomic polynomial factors of Borwein trinomials. We complement abstractions with lots of examples.

Paper Structure

This paper contains 13 sections, 37 theorems, 140 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Setting the Stage
  2. Trees
  3. Discrepancies
  4. Equivalence
  5. Rod Set Algebra
  6. Generating functions
  7. Duality and its Applications
  8. Rod Sets and Binomial Coefficients
  9. Periodicity
  10. Expandability and Scaling
  11. Fibonacci and Lucas 2-Expansions
  12. Borwein trinomials
  13. $2$-Expandability when $\max R > 2$

Key Result

Theorem 1.6

For rod set $R$, the net train count sequence $\mathop{\mathrm{F}}\nolimits(n,R)$ satisfies the recursion with initial conditions

Figures (7)

  • Figure 1: The $14$ trains of length $10$ made from Cuisenaire™ rods of lengths $2$ (red), $3$ (green), and $5$ (yellow). Each row is a separate train.
  • Figure 2: Converting the top train of $1$'s and $2$'s into the bottom train of $1$s, $3$s, and $4$s, with a $2$ at the end.
  • Figure 3: The first four levels of $\mathop{\mathrm{Tree}}\nolimits([1,{\color{red}\overline{2}}])$
  • Figure 4: Expansions of $[1,2]$ and $[1,{\color{red}\overline{2}}]$, with internal nodes circled, leaves boxed.
  • Figure 5: Three trees for the expansion $[1,2]\xrightarrow{\space[1,2,3]\space}[2,3,4,4,5]$.
  • ...and 2 more figures

Theorems & Definitions (59)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • ...and 49 more