• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1033-1048
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• 2020

We synthesize the recent work done on conditionally defined Lucas and Fibonacci numbers, tying together various definitions and results generalizing the linear recurrence relation. Allowing for any initial conditions, we determine the generating function and a Binet-like formula for the general sequence, in both the positive and negative directions, as well as relations among various sequence pairs. We also determine conditions for periodicity of these sequences and graph some recurrent figures in Python.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.549-559
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• 2018

In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {$U_{kn}$} and {$V_{kn}$}. For example, for an odd prime number p, $${\sum\limits_{i=1}^{p-1}}H_i{\frac{U_{k(i+m)}}{V^i_k}}{\equiv}{\frac{(-1)^kU_{k(m+1)}}{_pV^{p-1}_k}}(V^p_k-V_{kp})(mod\;p)$$, where $m{\in}{\mathbb{Z}}$ and $k{\in}{\mathbb{Z}}$ with $p{\nmid}V_k$.