[Paper Review] New Identities from a Combinatorial Approach to Generalized Fibonacci and Generalized Lucas Numbers

Robson da Silva|arXiv (Cornell University)|Dec 12, 2013

Advanced Mathematical Theories and Applications3 citations

TL;DR

This paper introduces a novel combinatorial tiling model for generalized Fibonacci and Lucas numbers using $1\times(n+1)$ boards tiled with $1\times1$ squares (white or black) and $1\times k$ gray rectangles, where exactly one black square appears in the first $k$ positions. The model yields new identities for generalized and classical Fibonacci and Lucas numbers, including explicit formulas involving nested sums of Fibonacci numbers and a new identity for $L_n$ in terms of $F_{n-5}$ and weighted sums of earlier Fibonacci numbers.

ABSTRACT

We present here some new identities for generalizations of Fibonacci and Lucas numbers by combinatorially interpreting these numbers in terms of numbers of certain tilings of a $1 imes m$ board. As a consequence, some new interesting identities involving the ordinaries Fibonacci and Lucas numbers are derived.

Motivation & Objective

• To develop a new combinatorial interpretation of generalized Fibonacci and Lucas numbers using tiling of a $1\times(n+1)$ board.
• To derive novel identities for generalized Fibonacci and Lucas numbers through tiling enumeration.
• To recover and extend known identities for classical Fibonacci and Lucas numbers by specializing the generalized framework to $k=2$.
• To provide combinatorial proofs that reveal structural insights beyond algebraic induction.

Proposed method

• Define $f(k,n)$ as the number of tilings of a $1\times(n+1)$ board with $1\times1$ white/black squares and $1\times k$ gray rectangles, with exactly one black square in the first $k$ positions.
• Establish recurrence $f(k,n) = f(k,n-1) + f(k,n-k)$ for $n \geq k$, matching the recurrence for $F(k,n)$, thus proving $F(k,n) = f(k,n)$.
• Extend the tiling model to include additional constraints for $L(k,n)$: at least $k-1$ white squares in the last $k$ positions when $n \geq 2k$, and minimum white square counts in tails of size $k-1$ or $2k-1$.
• Count tilings by classifying them based on the number of gray rectangles before the tail and the positions of the black and gray pieces.
• Use double summations over positions of white squares and gray rectangles to enumerate tilings with two or more rectangles before the tail.
• Specialize results to $k=2$ to derive new identities for classical Fibonacci and Lucas numbers.

Experimental results

Research questions

• RQ1 How can generalized Fibonacci and Lucas numbers be reinterpreted combinatorially via tiling models with constrained black square placement?
• RQ2 What new identities emerge from enumerating tilings with specific structural constraints on black and gray tiles?
• RQ3 Can the tiling model generate identities not derivable by standard recurrence substitution or induction?
• RQ4 How do the combinatorial structures for $L(k,n)$ differ from those of $F(k,n)$, especially in terms of tail constraints?
• RQ5 What new identities for classical Fibonacci and Lucas numbers arise when $k=2$ in the generalized tiling model?

Key findings

• The paper establishes $F(k,n) = f(k,n)$, where $f(k,n)$ counts tilings of a $1\times(n+1)$ board with exactly one black square in the first $k$ positions and any number of $1\times1$ white or $1\times k$ gray rectangles.
• For $n \geq 4k-1$, the generalized Lucas number $L(k,n)$ is given by a complex formula involving $k^2(n+4) - 3k^3 - \frac{k^2(k-1)}{2}$ plus double sums of $F(k,n-3k+1-i-j)$ and $(k-1)F(k,n-4k+1-i-j)$ over valid $i,j$ ranges.
• When specialized to $k=2$, the identity for $L(k,n)$ yields a new formula: $L_n = 4n - 10 + F_{n-5} + \sum_{i=1}^{n-5} 2i F_{n-5-i}$ for $n \geq 7$, expressing Lucas numbers as a linear combination of earlier Fibonacci numbers with increasing weights.
• The identity $L_n = 7 + \sum_{i=0}^{n-5} L_{n-2-i}$ is derived for $n \geq 5$, showing a recursive decomposition of Lucas numbers into earlier Lucas numbers starting from $L_3=4$.
• The tiling model provides a structural, combinatorial explanation for identities that are otherwise proven by induction, revealing deeper connections between tiling configurations and number sequences.
• The method successfully generates new identities not found in prior works, including those in [13], and offers a framework for discovering further identities through enumeration of constrained tiling patterns.