---
title: Gaussian Modified Third-Order Jacobsthal Numbers
url: https://www.emergentmind.com/topics/gaussian-modified-third-order-jacobsthal-numbers
type: topic
---

# Gaussian Modified Third-Order Jacobsthal Numbers

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arXiv search query: "Gaussian modified third-order Jacobsthal numbers"
Gaussian modified third-order Jacobsthal numbers are Gaussian-number-valued analogues of the modified third-order Jacobsthal sequence. In the formulation introduced by Morales, they arise as the special case \(a=3\), \(b=1\), \(c=3\) of the generalized Gaussian third-order Jacobsthal sequence and satisfy the third-order recurrence
\[
Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},
\]
with initial values
\[
Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.
\]
The same work records the relation
\[
Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},
\]
linking the Gaussian sequence to the real modified third-order Jacobsthal numbers \(K_n^{(3)}\). Closely related literature also studies complex or unrestricted complex variants of the same underlying recurrence, so the subject sits at the intersection of linear recurrences, periodic root-of-unity corrections, and hypercomplex lifts of Jacobsthal-type sequences [2508.11650].

## 1. Real precursor and emergence of the Gaussian sequence

The real precursor is the modified third-order Jacobsthal sequence \(\{K_n^{(3)}\}_{n\ge 0}\), introduced with recurrence
\[
K_{n+3}^{(3)}=K_{n+2}^{(3)}+K_{n+1}^{(3)}+2K_n^{(3)},
\]
and initial conditions
\[
K_0^{(3)}=3,\qquad K_1^{(3)}=1,\qquad K_2^{(3)}=3.
\]
Its initial segment is
\[
3,1,3,10,15,31,66,\dots,
\]
and it satisfies
\[
K_n^{(3)}=J_n^{(3)}+2J_{n-1}^{(3)},
\]
where \(J_n^{(3)}\) denotes the third-order Jacobsthal sequence [1905.00725].

A common misconception is that the Gaussian version was already defined in this 2019 note. It was not. The note develops the real modified sequence, its generating function, Binet-style formula, periodic correction terms, and a range of identities, thereby providing the algebraic template later used for Gaussian generalization. The explicit Gaussian modified third-order Jacobsthal numbers appear later as a specialization of generalized Gaussian third-order Jacobsthal numbers, with the real initial data \(a=3\), \(b=1\), \(c=3\) selected so that the resulting Gaussian sequence is the modified Gaussian analogue of the real sequence above [2508.11650].

This historical progression is mathematically natural. The real sequence already has a characteristic polynomial with one real root and two nonreal cube-root components, so the transition from integer-valued to Gaussian-valued recurrences preserves the same spectral structure.

## 2. Definition, initial values, and characteristic structure

In the generalized Gaussian framework, the sequence \(\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}\) is defined by
\[
\mathcal{J}g_{n+3}^{(3)}=\mathcal{J}g_{n+2}^{(3)}+\mathcal{J}g_{n+1}^{(3)}+2\mathcal{J}g_n^{(3)},
\]
with initial values
\[
\mathcal{J}g_0^{(3)}=a+\frac12(c-b-a)i,\qquad
\mathcal{J}g_1^{(3)}=b+ai,\qquad
\mathcal{J}g_2^{(3)}=c+bi.
\]
Specializing to \(a=3\), \(b=1\), \(c=3\) yields the Gaussian modified third-order Jacobsthal numbers:
\[
Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.
\]
The paper also states
\[
Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},\qquad n\ge 1.
\]
Accordingly, the first several terms are
\[
3-\frac12 i,\;1+3i,\;3+i,\;10+3i,\;15+10i,\;31+15i,\;66+31i,\;127+66i,\;255+127i,\dots
\]
[2508.11650].

The characteristic equation is
\[
\xi^3-\xi^2-\xi-2=0,
\]
with roots
\[
\xi=2,\qquad \omega_1,\qquad \omega_2,
\]
where \(\omega_1,\omega_2\) are the roots of
\[
x^2+x+1=0.
\]
These satisfy
\[
\omega_1+\omega_2=-1,\qquad \omega_1\omega_2=1,\qquad \omega_1-\omega_2=i\sqrt3.
\]
The oscillatory part is encoded by the \(3\)-periodic auxiliary sequence
\[
\Omega_n=\frac{1}{\omega_1-\omega_2}\left(\omega_1^n-\omega_2^n\right),
\]
for which
\[
\Omega_n=
\begin{cases}
0,& n\equiv 0 \pmod 3,\\
1,& n\equiv 1 \pmod 3,\\
-1,& n\equiv 2 \pmod 3.
\end{cases}
\]

This decomposition isolates two distinct behaviors: an exponential contribution controlled by the real root \(2\), and a bounded \(3\)-periodic correction controlled by the nonreal roots. That separation governs essentially all closed forms and identities for the sequence.

## 3. Closed form, generating function, and basic linear identities

The exact Binet-type formula for the Gaussian modified third-order Jacobsthal numbers is
\[
Kg_n^{(3)}=\left(1+\frac i2\right)2^n+(1+i)\Omega_n+(2-i)\Omega_{n+1}.
\]
The corresponding generating function is
\[
g\!\left(Kg_n^{(3)};\xi\right)=
\frac{3-2\xi-\xi^2-\frac12(1-7\xi+3\xi^2)i}{1-\xi-\xi^2-2\xi^3}.
\]
Both formulas are obtained by specializing the general Gaussian third-order Jacobsthal theory to \(a=3\), \(b=1\), \(c=3\) [2508.11650].

Several basic linear identities follow immediately from the same specialization. The shift-by-\(3\) identity is
\[
Kg_{n+3}^{(3)}=Kg_n^{(3)}+7\left(1+\frac i2\right)2^n,
\]
and the three-term sum identity is
\[
Kg_n^{(3)}+Kg_{n+1}^{(3)}+Kg_{n+2}^{(3)}=
7\left(1+\frac i2\right)2^n.
\]
The paper also gives a first-step relation:
\[
Kg_{n+1}^{(3)}=
2Kg_n^{(3)}-\left[(4-i)\Omega_n+(5-4i)\Omega_{n+1}\right].
\]

These formulas show that the Gaussian sequence inherits the real third-order Jacobsthal dynamics without altering the denominator of the generating function or the characteristic roots. The only change is the Gaussian-valued numerator data, which modifies the coefficients of the exponential and periodic terms.

## 4. Summation laws, negative indices, and determinant identities

The finite-sum formula is
\[
\sum_{l=0}^{n}Kg_l^{(3)}
=\frac13\left[Kg_{n+2}^{(3)}+2Kg_n^{(3)}-\frac32 i\right].
\]
For negative subscripts, the recurrence is extended by
\[
\mathcal{J}g_{-(n+3)}^{(3)}
=\frac12\left[-\mathcal{J}g_{-(n+2)}^{(3)}-\mathcal{J}g_{-(n+1)}^{(3)}+\mathcal{J}g_{-n}^{(3)}\right],
\]
and the specialized negative-index formulas are
\[
Kg_{-n}^{(3)}
=\left(1+\frac i2\right)\left(\frac12\right)^n-(1+i)\Omega_n-(2-i)\Omega_{n-1},
\]
equivalently,
\[
Kg_{-n}^{(3)}
=\left(1+\frac i2\right)\left(\frac12\right)^n-(1+2i)\Omega_n+(2-i)\Omega_{n+1}.
\]
These identities provide an explicit analytic continuation of the sequence to negative indices [2508.11650].

The same paper specializes general determinant-type identities. Writing
\[
\Theta_n=(4+i)\Omega_n+(5-4i)\Omega_{n+1},
\]
the specialized d’Ocagne identity is
\[
Kg_{m+1}^{(3)}Kg_n^{(3)}-Kg_m^{(3)}Kg_{n+1}^{(3)}
=
-3i\,\Omega_{n-m}
+\left(1+\frac i2\right)\left[2^m\Theta_n-2^n\Theta_m\right].
\]
Cassini’s identity becomes
\[
\left[Kg_n^{(3)}\right]^2-Kg_{n-1}^{(3)}Kg_{n+1}^{(3)}
=
-3i+\left(1+\frac i2\right)2^{n-1}
\left[(2+5i)\Omega_n+(13-2i)\Omega_{n+1}\right].
\]

These relations are the Gaussian counterparts of classical determinant identities for second- and higher-order recurrence sequences. Their structure again splits into an exponential part and a periodic root-of-unity part, which is characteristic of third-order Jacobsthal-type families.

## 5. Alternative complex formulations and normalization issues

A related but distinct line of work studies unrestricted complex modified third-order Jacobsthal numbers in the form
\[
K_n^{(3)}+K_{n+a}^{(3)}i,\qquad a\in\mathbb Z,
\]
with the “usual” complex specialization
\[
K_n^{(3)}+K_{n+1}^{(3)}i.
\]
In that setting, the sequence is treated as a specialization of unrestricted modified third-order Jacobsthal quaternions, and the complex sequence satisfies the same third-order recurrence
\[
G_{n+3}^{(a)}=G_{n+2}^{(a)}+G_{n+1}^{(a)}+2G_n^{(a)},
\]
where \(G_n^{(a)}=K_n^{(3)}+K_{n+a}^{(3)}i\) [2409.18975].

The corresponding Binet-like formula is
\[
K_n^{(3)}+K_{n+a}^{(3)}i
=
2^n(1+2^a i)+\omega_1^n(1+\omega_1^a i)+\omega_2^n(1+\omega_2^a i),
\]
and the generating function is
\[
\sum_{n=0}^{\infty} \bigl(K_n^{(3)}+K_{n+a}^{(3)}i\bigr)x^n
=
\frac{
(1+2^a i)(1+x+x^2)
+(1+\omega_1^a i)(1+(\omega_1-1)x+2\omega_2x^2)
+(1+\omega_2^a i)(1+(\omega_2-1)x+2\omega_1x^2)
}{1-x-x^2-2x^3}.
\]

This formulation is closely related to, but not identical with, Morales’s Gaussian modified sequence. In [2508.11650] the principal normalization is
\[
Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)}
\]
for \(n\ge 1\), together with the generalized initialization
\[
Kg_0^{(3)}=3-\frac12 i.
\]
In [2409.18975] the standard complex choice is instead
\[
K_n^{(3)}+K_{n+1}^{(3)}i,
\]
whose first terms begin
\[
3+i,\;1+3i,\;3+10i,\;10+15i,\dots
\]
The difference is therefore a matter of shift convention and initialization rather than recurrence structure.

A second normalization issue appears in quaternion literature. One 2025 paper uses a companion “modified third-order Jacobsthal” sequence with initial values
\[
K_0^{(3)}=0,\qquad K_1^{(3)}=1,\qquad K_2^{(3)}=3,
\]
while still retaining the same recurrence and characteristic polynomial [2504.03646]. This shows that the term “modified third-order Jacobsthal” is not completely uniform across the literature. For Gaussian modified third-order Jacobsthal numbers in the sense of Morales, however, the underlying real sequence is the one with initial values \(3,1,3\).

## 6. Hypercomplex extensions and broader structural context

The Gaussian sequence belongs to a larger program of lifting third-order Jacobsthal-type recurrences into hypercomplex algebras. One such development introduces third-order Jacobsthal \(3\)-parameter generalized quaternions and their modified counterparts by embedding consecutive scalar terms into quaternion coordinates:
\[
JQ_n^{(3)} = J_n^{(3)} + J_{n+1}^{(3)}e_1 + J_{n+2}^{(3)}e_2 + J_{n+3}^{(3)}e_3,
\]
\[
KQ_n^{(3)} = K_n^{(3)} + K_{n+1}^{(3)}e_1 + K_{n+2}^{(3)}e_2 + K_{n+3}^{(3)}e_3.
\]
These quaternion sequences satisfy the same recurrence componentwise and admit Binet formulas, generating functions, sum identities, Cassini identities, and d’Ocagne/Vajda-type relations [2504.03646].

Earlier work on third-order Jacobsthal quaternions develops the same recurrence backbone using the companion matrix
\[
M=
\begin{bmatrix}
1&1&2\\
1&0&0\\
0&1&0
\end{bmatrix},
\]
together with quaternion generating functions, Binet formulas, and matrix representations [1706.08989]. This suggests that the Gaussian modified sequence is best understood not as an isolated construction, but as one instance of a recurrence-preserving lift from \(\mathbb Z\) to a richer coefficient domain.

From this perspective, the central invariant is the scalar recurrence
\[
x_{n+3}=x_{n+2}+x_{n+1}+2x_n,
\]
with characteristic polynomial
\[
r^3-r^2-r-2=(r-2)(r^2+r+1).
\]
Once that recurrence is fixed, integer, Gaussian, complex, quaternionic, and generalized quaternionic versions differ primarily in their initial data and ambient algebra. The repeated appearance of the same denominator
\[
1-x-x^2-2x^3
\]
in generating functions, and the same decomposition into a \(2^n\) term plus a \(3\)-periodic correction, is the unifying feature across these families.

In that sense, Gaussian modified third-order Jacobsthal numbers occupy a precise intermediate position. They retain the scalar third-order Jacobsthal spectral data, but package it in Gaussian form; they are more structured than a purely formal complexification, yet remain commutative and considerably simpler than quaternionic generalizations.

Source: https://www.emergentmind.com/topics/gaussian-modified-third-order-jacobsthal-numbers