# On the structure of periodic complex Horadam orbits

Journal article

*Carpathian Journal of Mathematics.*

Authors | Bagdasar, Ovidiu D., Larcombe, Peter J., Anjum, Ashiq and Bagdasar, O. |
---|---|

Abstract | Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat’s spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures. Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane. Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multisymmetric patterns can be recovered for selected parameter values. Some applications are also suggested. |

Numerous geometric patterns identified in nature, art or science can be generated from recurrent | |

Keywords | Discrete mathematics; Applied mathematics |

Year | 2016 |

Journal | Carpathian Journal of Mathematics |

Publisher | North University of Baia Mare (Romania) |

ISSN | 1584-2851 |

1843-4401 | |

Web address (URL) | http://carpathian.ubm.ro/?m=past_issues&issueno=Vol.%2032%20(2016),%20No.%201 |

https://publons.com/wos-op/publon/14541647/ | |

Output status | Published |

Publication dates | 2016 |

Publication process dates | |

Deposited | 16 Nov 2016, 13:28 |

Contributors | University of Derby |

File | File Access Level Restricted |

File | File Access Level Restricted |

File | File Access Level Open |

https://repository.derby.ac.uk/item/937yv/on-the-structure-of-periodic-complex-horadam-orbits

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Lowndes, Val, Bagdasar, Ovidiu and Berry, Stuart 2017. Case studies: Using heuristics. in: Springer.##### Further use of heuristic methods.

Lowndes, Val, Berry, Stuart, Parkes, Chris, Bagdasar, Ovidiu and Popovici, Nicolae 2017. Further use of heuristic methods. in: Springer.##### On the geometry of certain periodic non-homogeneous Horadam sequences.

Bagdasar, Ovidiu and Popa, Ioan-Lucian 2016. On the geometry of certain periodic non-homogeneous Horadam sequences.*Electronic Notes in Discrete Mathematics.*https://doi.org/10.1016/j.endm.2016.11.002

##### A closed form formulation for the general term of a scaled triple power product recurrence sequence.

Larcombe, Peter J. and Fennessey, Eric J. 2017. A closed form formulation for the general term of a scaled triple power product recurrence sequence.*Fibonacci Quarterly.*

##### New results and conjectures on 2-partitions of multisets.

Bagdasar, Ovidiu and Andrica, Dorin 2017. New results and conjectures on 2-partitions of multisets.*Proceedings of the 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO).*https://doi.org/10.1109/ICMSAO.2017.7934928

##### Opinions, opinions, opinions ...

Larcombe, Peter J. 2017. Opinions, opinions, opinions ...*Mathematics Today.*

##### Mathematics as a mirror of painting.

Larcombe, Peter J. 2017. Mathematics as a mirror of painting.*Mathematics Today.*

##### Horadam sequences: A survey update and extension.

Larcombe, Peter J. 2017. Horadam sequences: A survey update and extension.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### On sequence-based closed form entries for an exponentiated general $2 \times 2$ matrix: A re-formulation and an application.

Larcombe, Peter J. and Fennessey, Eric J. 2017. On sequence-based closed form entries for an exponentiated general $2 \times 2$ matrix: A re-formulation and an application.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### Qualitative case study methodology: Automatic design and correction of ceramic colors

Bagdasar, O., Birluţiu, A., Chen, M. and Popa, I.-L. 2017. Qualitative case study methodology: Automatic design and correction of ceramic colors. IEEE. https://doi.org/10.1109/icstcc.2017.8107118##### New results and conjectures on 2-partitions of multisets

Bagdasar, O. and Andrica, D. 2017. New results and conjectures on 2-partitions of multisets.*7th International Conference on Modeling, Simulation, and Applied Optimization, ICMSAO 2017.*IEEE Computer Society. https://doi.org/10.1109/icmsao.2017.7934928

##### ON THE MASKED PERIODICITY OF HORADAM SEQUENCES: A GENERATOR-BASED APPROACH

Bagdasar, Ovidiu D., Larcombe, Peter J. and Bagdasar, O. 2017. ON THE MASKED PERIODICITY OF HORADAM SEQUENCES: A GENERATOR-BASED APPROACH.*Fibonacci Quarterly.*

##### LOCAL MAXIMIZERS OF GENERALIZED CONVEX VECTOR-VALUED FUNCTIONS

Bagdasar, Ovidiu, Popovici, Nicolae and Bagdasar, O. 2017. LOCAL MAXIMIZERS OF GENERALIZED CONVEX VECTOR-VALUED FUNCTIONS.*Journal of Nonlinear and Convex Analysis.*

##### Conditions governing cross-family member equality in a particular class of polynomial families

Larcombe, Peter J. and Fennessey, Eric J. 2014. Conditions governing cross-family member equality in a particular class of polynomial families.*Fibonacci Quarterly.*

##### On certain series expansions of the sine function: Catalan numbers and convergence

Larcombe, Peter J., O'Neill, Sam T. and Fennessey, Eric J. 2014. On certain series expansions of the sine function: Catalan numbers and convergence.*Fibonacci Quarterly.*

##### On the number of complex horadam sequences with a fixed period

Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the number of complex horadam sequences with a fixed period.*Fibonacci Quarterly.*

##### On the characterization of periodic complex Horadam sequences

Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the characterization of periodic complex Horadam sequences.*Fibonacci Quarterly.*

##### On a result of Bunder involving Horadam sequences: A proof and generalization

Larcombe, Peter J. and Bagdasar, Ovidiu 2013. On a result of Bunder involving Horadam sequences: A proof and generalization.*Fibonacci Quarterly.*

##### Influence discovery in semantic networks: An initial approach

Trovati, M. and Bagdasar, O. 2014. Influence discovery in semantic networks: An initial approach. IEEE. https://doi.org/10.1109/uksim.2014.48##### Complementary iInequalities involving the Stolarsky Mean

Bagdasar, Ovidiu 2010. Complementary iInequalities involving the Stolarsky Mean.*Journal of Inequalities and Applications.*https://doi.org/10.1155/2010/492570

##### Concise computer mathematics: tutorials on theory and problems

Bagdasar, Ovidiu 2013.*Concise computer mathematics: tutorials on theory and problems.*Springer.

##### On the characterization of periodic generalized Horadam sequences

Bagdasar, Ovidiu and Larcombe, Peter J. 2014. On the characterization of periodic generalized Horadam sequences.*Journal of Difference Equations and Applications.*https://doi.org/10.1080/10236198.2014.891022

##### Local maximum points of explicitly quasiconvex functions

Bagdasar, Ovidiu and Popovici, Nicolae 2014. Local maximum points of explicitly quasiconvex functions.*Optimization Letters.*https://doi.org/10.1007/s11590-014-0781-3

##### A horadam-based pseudo-random number generator

Bagdasar, O.D., Chen, M. and Bagdasar, O. 2014. A horadam-based pseudo-random number generator. IEEE. https://doi.org/10.1109/uksim.2014.55##### Alwyn Francis Horadam, 1923-2016: A personal tribute to the man and his sequence

Larcombe, Peter J. 2016. Alwyn Francis Horadam, 1923-2016: A personal tribute to the man and his sequence.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### A scaled power product recurrence examined using matrix methods

Larcombe, Peter J. and Fennessey, Eric J. 2016. A scaled power product recurrence examined using matrix methods.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach

Larcombe, Peter J. 2016. On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach.*Fibonacci Quarterly.*

##### On a scaled balanced-power product recurrence

Larcombe, Peter J. and Fennessey, Eric J. 2016. On a scaled balanced-power product recurrence.*Fibonacci Quarterly.*

##### A short monograph on exposition and the emotive nature of research and publishing

Larcombe, Peter J. 2016. A short monograph on exposition and the emotive nature of research and publishing.*Mathematics Today.*

##### A polynomial based construction of periodic Horadam sequences

Larcombe, Peter J. and Fennessey, Eric J. 2016. A polynomial based construction of periodic Horadam sequences.*Utilitas Mathematica.*

##### A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power

Larcombe, Peter J. 2016. A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### On the jacobsthal, horadam and geometric mean sequences

Larcombe, Peter J. and Rabago, Julius, F. T. 2016. On the jacobsthal, horadam and geometric mean sequences.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs

Larcombe, Peter J. 2015. A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs.*Fibonacci Quarterly.*

##### Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations

Larcombe, Peter J. 2015. Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations.*Fibonacci Quarterly.*

##### Closed form evaluations of some series involving Catalan numbers

Larcombe, Peter J. 2014. Closed form evaluations of some series involving Catalan numbers.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families

Larcombe, Peter J. and Fennessey, Eric J. 2015. A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families.*Fibonacci Quarterly.*

##### On the phenomenon of masked periodic Horadam sequences

Larcombe, Peter J. and Fennessey, Eric J. 2015. On the phenomenon of masked periodic Horadam sequences.*Utilitas Mathematica.*

##### On horadam sequence periodicity: A new approach

Larcombe, Peter J. and Fennessey, Eric J. 2015. On horadam sequence periodicity: A new approach.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### A non-linear identity for a particular class of polynomial families

Larcombe, Peter J. and Fennessey, Eric J. 2014. A non-linear identity for a particular class of polynomial families.*Fibonacci Quarterly.*

##### On cyclicity and density of some Catalan polynomial sequences

Larcombe, Peter J. and Fennessey, Eric J. 2014. On cyclicity and density of some Catalan polynomial sequences.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### Some factorisation and divisibility properties of Catalan polynomials

Jarvis, Frazer A., Larcombe, Peter J. and Fennessey, Eric J. 2014. Some factorisation and divisibility properties of Catalan polynomials.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### Generalised Catalan polynomials and their properties

Larcombe, Peter J., Jarvis, Frazer A. and Fennessey, Eric J. 2014. Generalised Catalan polynomials and their properties.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs

Larcombe, Peter J., Kirschenhofer, Peter and Fennessey, Eric J. 2014. The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs.*Utilitas Mathematica.*

##### Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems

Popa, I.-L., Ceauşu, T., Bagdasar, O. and Agarwal, R.P. 2016. Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems.*Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica.*27 (2), pp. 153-166. https://doi.org/10.2478/auom-2019-0025

##### On some functions involving the lcm and gcd of integer tuples

Bagdasar, Ovidiu 2014. On some functions involving the lcm and gcd of integer tuples.*Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics.*https://doi.org/10.5937/SPSUNP1402091B

##### ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF

Larcombe, Peter J., Bagdasar, O. and Fennessey, Eric J. 2014. ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF.*Fibonacci Quarterly.*pp. 175-177.