TITLE:
Higher-Order k-Nacci Quasigeometric Systems
AUTHORS:
Lovemore Mamombe
KEYWORDS:
Quasigeometric Sequence, k-Nacci Sequence, Multinacci Constant, Nearest-Integer Rounding, Parent Number, Barren Integer, Beatty-Type Sequence, Discrepancy, Symbolic Dynamics
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.16 No.4,
October
15,
2026
ABSTRACT: We herein introduce and develop a general theory of higher-order k-nacci quasigeometric systems by imposing simultaneously the k-step additive recurrence
h
n+k
=
h
n+k−1
+
h
n+k−2
+⋯+
h
n
and nearest-integer multiplication by the dominant k-nacci root
α
k
,
h
n+1
=rnd(
α
k
h
n
)
,
α
k
k
=
α
k
k−1
+⋯+
α
k
+1
. For the rounded orbit
x
0
=n
and
x
j+1
=rnd(
α
k
x
j
)
, admissibility is measured by the finite k-step discrepancy
D
k
(
n
)=
x
k
−
∑
j=0
k−1
x
j
. Using the exact root identity
α
k
+
α
k
−k
=2
, the rounded dynamics reduce to the integer correction map
R
k
(
n
)=2n−
c
k
(
n
)
,
c
k
(
n
)=⌊
α
k
−k
n+1/2
⌋
, which yields the exact correction-sum formula
D
k
(
n
)=n−
∑
j=0
k−1
c
k
(
x
j
)
. Hence n is a genuine k-nacci parent number precisely when
D
k
(
n
)=0
. The same representation gives k-step recovery and discrepancy conservation,
rnd(
R
k
k
(
n
)
α
k
k
)=n
,
D
k
(
R
k
(
n
)
)=
D
k
(
n
)
. Consequently, barrenness is inherited along forward rounded orbits. Primitive barren integers are the barren members of the explicit no-predecessor sequence
q
k
(
m
)=⌊
[
α
k
/
(
α
k
−1
)
]m−1/
[
2(
α
k
−1
) ]
⌋
,
m≥1
, and every barren integer belongs to a unique forward orbit rooted at one such primitive generator. The formulas hold for arbitrary fixed k ≥ 3; Tribonacci, Tetranacci, Pentanacci, and higher-order examples illustrate the general theory.