• The Pure and Applied Mathematics
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• v.31 no.4
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• pp.427-437
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• 2024

In this paper, we introduce the Lucas-Padovan quaternions sequence. Initiating the studies based on the Padovan quaternion coefficients in relation to their recurrence, their matrix representation is then defined. We investigate various aspects of these quaternions, including summation formulas and binomial sums.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.977-988
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• 2022

The Padovan sequence is the third-order linear recurrence (𝓟_n)_n≥0 defined by 𝓟_n = 𝓟_n-2 + 𝓟_n-3 for all n ≥ 3 with initial conditions 𝓟₀ = 0 and 𝓟₁ = 𝓟₂ = 1. In this paper, we investigate a generalization of the Padovan sequence called the k-generalized Padovan sequence which is generated by a linear recurrence sequence of order k ≥ 3. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences.

• Honam Mathematical Journal
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• v.46 no.4
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• pp.552-566
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• 2024

In this paper, we investigate the perfect powers that are the difference between Perrin numbers (R_k)_k≥0 and Padovan numbers (P_k)_k≥0. Hence, we solve the equations P_n = x^a, 2P_n = x^a, P_n - R_m = x^a, or R_n - P_m = x^a such that a ≥ 1 and 2 ≤ x ≤ 10 are positive integers and n, m, and k are non-negative integers.