In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

In this study, the $k$-Oresme and $k$-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the $k$-Oresme and $k$-Oresme-Lucas sequences are presented. In addition, we give these sequences the Binet formulas, generating functions, Cassini identity, Catalan identity etc. Moreover, the $k$-Oresme and $k$-Oresme-Lucas sequences are associated with Fibonacci, Pell numbers and Lucas, and Pell- Lucas numbers, respectively. Finally, the Catalan transforms of these sequences are given and Hankel transforms are applied to these Catalan sequences and associated with the terms of the sequence.

In this study, we define the Bronze Leonardo, Bronze Leonardo–Lucas, and Modified Bronze Leonardo sequences, and some terms of these sequences are given. Then, we give special summation formulas, special generating functions, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence’s generating functions, and the third is with the help of the matrices. In addition, we find the special relations between the terms of the Bronze Leonardo, Bronze Leonardo–Lucas, and Modified Bronze Leonardo sequences. Moreover, we examine the relationships among the Bronze Fibonacci and Bronze Lucas sequences of these sequences. Finally, we associate these sequences with the matrices.

In this study, we establish some properties of Bronze Fibonacci and Bronze Lucas sequences. Then we find the relationships between the roots of the characteristic equation of these sequences with these sequences. What is interesting here is that even though the roots change, equality is still maintained. Also, we derive the special relations between the terms of these sequences. We give the important relations among these sequences, positive and negative index terms, with the sum of the squares of two consecutive terms being related to these sequences. In addition, we present the application of generalized Bronze Fibonacci sequences to hyperbolic quaternions. For these hyperbolic quaternions, we give the summation formulas, generating functions, etc. Moreover, we obtain the Binet formulas in two different ways. The first is in the known classical way and the second is with the help of the sequence’s generating functions. In addition, we calculate the special identities of these hyperbolic quaternions. Furthermore, we examine the relationships between the hyperbolic Bronze Fibonacci and Bronze Lucas quaternions. Finally, the terms of the generalized Bronze Fibonacci sequences are associated with their hyperbolic quaternion values.

Recursive sequences with gaps have been studied previously. This paper considers some elementary properties of such sequences where the gaps have been created on a regular basis from sequence to sequence – ‘discatenated’ (systematic gaps) and ‘lacunary’ (general gaps). In particular, their generating functions are developed in order to open up their general terms and relations with other properties.

In this study, we define the $k$-Cullen, $k$-Cullen–Lucas, and Modified $k$-Cullen sequences, and certain terms in these sequences are given. Then, we obtain the Binet formulas, generating functions, summation formulas, etc. In addition, we examine the relations among the terms of the $k$-Cullen, $k$-Cullen–Lucas, Modified $k$-Cullen, Cullen, Cullen–Lucas, Modified Cullen, $k$-Woodall, $k$-Woodall–Lucas, Modified $k$-Woodall, Woodall, Woodall–Lucas, and Modified Woodall sequences. The generating functions were derived and analyzed, especially for cases where Fibonacci numbers were assigned to parameter $k.$ Graphical representations of the generating functions and their logarithmic transformations revealed interesting growth trends and convergence behavior. Further, by multiplying the generating functions with exponential expressions such as $e^k$, we explored the self-similar nature and mirrored dynamics among the sequences. Specifically, it was observed that the Modified Cullen sequence exhibited a symmetric and inverse-like resemblance to the Cullen and Cullen–Lucas sequences, suggesting the presence of deeper structural dualities. Additionally, indefinite integrals of the generating functions were computed and visualized over a range of Fibonacci-indexed k values. These integral-based graphs further reinforced the phenomenon of symmetry and self-similarity, particularly in the Modified Cullen sequence. A key insight of this study is the discovery of a structural duality between the Modified Cullen and standard Cullen-type sequences, supported both algebraically and graphically. This duality suggests new avenues for analyzing generalized recursive sequences through generating function transformations. This observation provides new insight into the structural behavior of generalized Cullen-type sequences.