Structural and self-similarity properties of higher-order Horadam numbers

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Kuloğlu B., Peters J. F., Uysal M., Özkan E.

8. INTERNATIONAL CONFERENCE ON ADVANCES IN NATURAL AND APPLIED SCIENCES, Antalya, Türkiye, 28 - 31 Ekim 2025, ss.107, (Özet Bildiri)

  • Yayın Türü: Bildiri / Özet Bildiri
  • Basıldığı Şehir: Antalya
  • Basıldığı Ülke: Türkiye
  • Sayfa Sayıları: ss.107
  • Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
  • Erzincan Binali Yıldırım Üniversitesi Adresli: Evet

Özet

In this study, we introduce and investigate higher-order Horadam numbers as a natural

generalization of classical number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, and

Mersenne. We derive their fundamental properties including recurrence relations, Binet-

type formulas, generating functions, summation formulas, and matrix representations.

Furthermore, we explore Catalan transforms of higher-order Horadam numbers and

analyze their structural behaviors. Particular emphasis is placed on the emergence of self-

similarity phenomena in nonlinear graphs constructed from the generating functions of

these sequences. The study highlights how self-similarity, inherent in higher-order

Horadam numbers, provides insights into infinite complexity within finite structures and

reveals potential applications in fields such as fractal compression, financial modeling, and

biological sequence analysis. Our results offer a comprehensive framework for

understanding higher-order Horadam numbers and open new perspectives for future

theoretical and applied research.