Tribonacci-Lucas Sequence

• The Tribonacci-Lucas sequence is a Lucas-type sequence defined by the third-order recurrence with initial values 3, 1, and 3, featuring a Binet-type representation.
• Analytic studies reveal its rational generating functions, characteristic roots, and matrix as well as hypercomplex realizations, aiding deeper insights in recurrence theory.
• Its close connections with Tribonacci numbers and polynomial analogues lead to practical identities and generalized extensions useful in spectral and Diophantine research.

The Tribonacci-Lucas sequence is the Lucas-type companion to the Tribonacci sequence: it satisfies the same third-order linear recurrence as the Tribonacci numbers, but with different initial data. In the standard normalization used across much of the recent literature, it is defined by

$$K_n=K_{n-1}+K_{n-2}+K_{n-3}\qquad (n\ge 3),\qquad K_0=3,\ K_1=1,\ K_2=3,$$

and begins

$$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$

The sequence admits a Binet-type representation, rational generating functions, negative-index extensions, polynomial and incomplete analogues, matrix and hypercomplex lifts, and generalized parameter families; it also appears in recent Diophantine classification problems and in matrix-theoretic constructions (Ddamulira, 7 Sep 2025).

1. Standard definition, characteristic roots, and analytic forms

The standard Tribonacci-Lucas sequence is governed by the characteristic polynomial

$x^3-x^2-x-1=0.$

If $\alpha,\beta,\gamma$ are its roots, then the standard Binet-type formula is

$K_n=\alpha^n+\beta^n+\gamma^n.$

In the arithmetic treatment of palindromic values, the real root $\alpha$ is singled out, with

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$

and the deviation from the dominant term is written as

$$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$

That same paper proves the useful size estimate

$\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$

where the notation $S_n$ is used for the standard Tribonacci-Lucas sequence rather than $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$0 (Ddamulira, 7 Sep 2025).

The ordinary generating function in the standard normalization is

$$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$1

This places the sequence within the standard rational-function framework of third-order linear recurrences and is repeatedly used as the entry point for matrix, polynomial, and transform constructions (Soykan, 2018).

A basic terminological point is that the phrase “Tribonacci-Lucas sequence” is not used with complete uniformity across the literature. Most papers in this corpus use the standard normalization $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$2, $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$3, $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$4, but generalized studies also introduce shifted or parameterized versions with different initial values and notation. This normalization issue is mathematically significant because it changes indexing conventions and the precise form of closed formulas.

2. Relations with Tribonacci numbers, addition laws, and negative indices

The Tribonacci-Lucas numbers are tightly linked to the Tribonacci sequence $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$5, defined by

$$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$6

Several identities express $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$7 as a fixed linear combination of nearby Tribonacci numbers: $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$8

$$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$9

$x^3-x^2-x-1=0.$0

These formulas show that the Tribonacci-Lucas sequence is not merely analogous to the Tribonacci sequence; it is embedded in the same cubic recurrence module and can be recovered from short Tribonacci windows (Soykan, 2018).

A particularly compact bridge identity is

$x^3-x^2-x-1=0.$1

This relation arises in the framework of generalized Tribonacci addition formulas and is highlighted as “a simple relation connecting the Tribonacci numbers and the Tribonacci-Lucas numbers” (Adegoke et al., 2018).

Summation identities are also explicit. For positive indices,

$x^3-x^2-x-1=0.$2

$x^3-x^2-x-1=0.$3

$x^3-x^2-x-1=0.$4

For negative indices,

$x^3-x^2-x-1=0.$5

$x^3-x^2-x-1=0.$6

$x^3-x^2-x-1=0.$7

These formulas are obtained as specializations of general summation theorems for generalized Tribonacci sequences (Soykan, 2019).

The negative-index extension itself is defined by reversing the recurrence: $x^3-x^2-x-1=0.$8 and it preserves the Binet form in the reciprocal powers,

$x^3-x^2-x-1=0.$9

Its generating function is

$\alpha,\beta,\gamma$0

This reversed theory is central in the matrix-sequence treatment, where negative indices become inverse-like objects rather than merely formal continuations (Soykan, 2018).

3. Polynomial, incomplete, and transform variants

A standard polynomial generalization replaces the scalar recurrence by

$\alpha,\beta,\gamma$1

with initial values

$\alpha,\beta,\gamma$2

The specialization

$\alpha,\beta,\gamma$3

recovers the ordinary Tribonacci-Lucas numbers. The corresponding generating function is

$\alpha,\beta,\gamma$4

and the polynomial sequence satisfies the identity

$\alpha,\beta,\gamma$5

with scalar specialization

$\alpha,\beta,\gamma$6

A further binomial-sum formula is

$\alpha,\beta,\gamma$7

which reduces at $\alpha,\beta,\gamma$8 to

$\alpha,\beta,\gamma$9

These results place the Tribonacci-Lucas sequence within a polynomial framework parallel to that of Lucas polynomials in the second-order setting (Kose et al., 2014).

A different extension introduces incomplete Tribonacci-Lucas numbers and polynomials by truncating Pascal-like triangle sums. For $K_n=\alpha^n+\beta^n+\gamma^n.$0,

$K_n=\alpha^n+\beta^n+\gamma^n.$1

with explicit form

$K_n=\alpha^n+\beta^n+\gamma^n.$2

Setting $K_n=\alpha^n+\beta^n+\gamma^n.$3 yields the incomplete Tribonacci-Lucas numbers $K_n=\alpha^n+\beta^n+\gamma^n.$4. These objects satisfy both homogeneous and non-homogeneous recurrences, for example

$K_n=\alpha^n+\beta^n+\gamma^n.$5

and they are related to incomplete Tribonacci polynomials by

$K_n=\alpha^n+\beta^n+\gamma^n.$6

Their generating functions are given in terms of the incomplete Tribonacci generating functions $K_n=\alpha^n+\beta^n+\gamma^n.$7 by

$K_n=\alpha^n+\beta^n+\gamma^n.$8

The incomplete theory shows that partial triangle sums preserve much of the cubic structure while introducing a truncation parameter $K_n=\alpha^n+\beta^n+\gamma^n.$9 (Yilmaz et al., 2014).

A separate transform line studies binomial transforms: one paper states that it applies “the binomial transforms to Tribonacci and Tribonacci-Lucas sequences” and that “the Binet formulas, summations, generating functions of these transforms are found using recurrence relations” (Yilmaz et al., 2014). Within the present corpus, the abstract establishes that such a transform theory exists, even though no further formulas are supplied here.

4. Matrix and hypercomplex realizations

The sequence has been lifted into several non-scalar settings in which consecutive terms are assembled into structured algebraic objects.

Realization	Defining object	Structural feature
Matrix sequence	$\alpha$0 matrices $\alpha$1	Same third-order recurrence
Matrix sequence with negative subscripts	$\alpha$2 matrices $\alpha$3	Backward recurrence and inverse-type behavior
Quaternion polynomials	$\alpha$4	Four consecutive components
Sedenions	$\alpha$5	Sixteen consecutive components

In the matrix setting, a Tribonacci-Lucas matrix sequence $\alpha$6 is defined recursively by the same third-order law as the scalar sequence. One explicit representation is

$\alpha$7

and the matrix Binet formula has the form

$\alpha$8

These matrices satisfy multiplicative identities such as

$\alpha$9

and

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$0

with explicit reductions to linear combinations of Tribonacci matrices. The resulting formalism converts additive index shifts into matrix multiplication (Soykan, 2018).

The negative-index matrix theory extends this construction to all integers. The backward recurrence is

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$1

and the corresponding Binet form is

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$2

In that framework, the Tribonacci matrices satisfy

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$3

and the Tribonacci-Lucas matrices inherit analogous multiplicative compatibility with positive and negative indices (Soykan, 2018).

Quaternion and sedenion lifts package finite windows of consecutive Tribonacci-Lucas values into hypercomplex elements. For Tribonacci-Lucas quaternion polynomials,

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$4

and

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$5

The ordinary generating function is given explicitly as a quaternion-valued rational function with denominator $$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$6, and setting $$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$7 recovers the classical Tribonacci-Lucas numbers in quaternion form (Cerda-Morales, 2017).

For sedenions,

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$8

defines a 16-dimensional Cayley-Dickson lift of the scalar sequence. The lifted Binet formula is

$$1.83<\alpha<1.84,\qquad |\beta|=|\gamma|=\alpha^{-1/2}<0.74,$$9

where $$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$0 are interpreted as sedenion-valued sums of powers of the scalar roots. The norm is

$$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$1

This construction preserves the cubic recurrence while embedding the sequence into a non-commutative, non-associative algebra with zero divisors (Soykan, 2018).

5. Generalized Tribonacci-Lucas families and circulant matrices

A parameterized generalization is introduced by the recurrence

$$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$2

with initial values

$$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$3

If $$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$4 are the three distinct real roots of the corresponding characteristic equation, then

$$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$5

When

$$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$6

this paper states that $$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$7 becomes the classical Tribonacci-Lucas sequence $$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$8 in that paper’s normalization. This normalization differs from the standard $$\xi(n):=S_n-\alpha^n=\beta^n+\gamma^n,\qquad |\xi(n)|\le \frac{2}{\alpha^{n/2}}.$$9, $\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$0, $\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$1 convention used elsewhere in the literature represented here (Yilmaz et al., 2014).

The same paper uses these generalized numbers to build the circulant matrix

$\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$2

whose first row is $\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$3. The eigenvalues are given explicitly by

$\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$4

for

$\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$5

Because $\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$6 is normal, its spectral norm is

$\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$7

In the special case $\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$8, this reduces to

$\alpha^{m-1}\le S_m<\alpha^{m+1}\qquad (m\ge 1),$9

The determinant is also written in closed factored form in terms of $S_n$0, $S_n$1, $S_n$2, and the characteristic-root combination

$S_n$3

This line of work is significant because it shifts the focus from scalar recurrence properties to spectral invariants of structured matrices built from Tribonacci-Lucas-type data (Yilmaz et al., 2014).

6. Arithmetic questions, research boundaries, and adjacent directions

A recent arithmetic study classifies those Tribonacci-Lucas numbers whose decimal expansions are palindromic concatenations of two distinct repdigits. The equation is written as

$S_n$4

and the main theorem proves that the only solution is

$S_n$5

Equivalently, the only Tribonacci-Lucas number of that form is

$S_n$6

with $S_n$7, $S_n$8, $S_n$9, and $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$00. The proof combines Baker-type lower bounds for linear forms in logarithms, continued-fraction reduction via Dujella–Pethő, an LLL step in an exceptional case, and a SageMath search for $$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$01 (Ddamulira, 7 Sep 2025).

The current literature also makes clear what should not be conflated with the Tribonacci-Lucas sequence. One paper on aperiodic monotile supertiles is governed by the second-order recurrence

$$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$02

and by Fibonacci and Lucas sequences, together with a Lucas-derived sequence

$$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$03

That work explicitly does not introduce a Tribonacci-Lucas sequence in the usual three-term-order sense (Dong, 2024).

Likewise, an infinite family of lacunary recurrences is proved for the classical Lucas numbers,

$$3,\,1,\,3,\,7,\,11,\,21,\,39,\,71,\,131,\,241,\,443,\,815,\,1499,\ldots$$04

and then iterated to a broader lacunary family. That paper does not mention Tribonacci-Lucas numbers explicitly, but it suggests a possible extension strategy once a suitable combinatorial model for third-order Lucas-type sequences is found. This suggests a methodological, rather than formulaic, connection to Tribonacci-Lucas theory (Mahanta et al., 2020).

A related caution applies to convolution theory. A study of convolution identities for Tribonacci numbers does not define the Tribonacci-Lucas sequence explicitly, but it develops a symmetric-root method showing that cubic-order convolutions typically resolve not into a single companion sequence, but into several generalized Tribonacci-type sequences with different initial conditions. A plausible implication is that Tribonacci-Lucas convolution identities should likewise be expected to involve a larger companion family than the Fibonacci/Lucas pair familiar from second-order theory (Komatsu et al., 2016).

Taken together, these results show that the Tribonacci-Lucas sequence occupies a stable core position in third-order linear recurrence theory: it is simultaneously a classical scalar sequence, a source of structured polynomial and matrix families, an input to hypercomplex lifts, a testing ground for Diophantine methods, and a reference point for broader generalized and analogical constructions.