Browse > Article
http://dx.doi.org/10.4134/CKMS.c210367

GENERALIZED PADOVAN SEQUENCES  

Bravo, Jhon J. (Departamento de Matematicas Universidad del Cauca)
Herrera, Jose L. (Departamento de Matematicas Universidad del Cauca)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.4, 2022 , pp. 977-988 More about this Journal
Abstract
The Padovan sequence is the third-order linear recurrence (𝓟n)n≥0 defined by 𝓟n = 𝓟n-2 + 𝓟n-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.
Keywords
Fibonacci number; generalized Padovan number; recurrence sequence;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 G. P. B. Dresden and Z. Du, A simplified Binet formula for k-generalized Fibonacci numbers, J. Integer Seq. 17 (2014), no. 4, Article 14.4.7, 9 pp.
2 A. Dubickas, K. G. Hare, and J. Jankauskas, No two non-real conjugates of a Pisot number have the same imaginary part, Math. Comp. 86 (2017), no. 304, 935-950. https://doi.org/10.1090/mcom/3103   DOI
3 A. C. Garcia Lomeli and S. Hernandez Hernandez, Powers of two as sums of two Padovan numbers, Integers 18 (2018), Paper No. A84, 11 pp.
4 A. C. Garcia Lomeli and S. Hernandez Hernandez, Repdigits as sums of two Padovan numbers, J. Integer Seq. 22 (2019), no. 2, art. 19.2.3, 10 pp.
5 A. C. Garcia Lomeli and S. Hernandez Hernandez, Pillai's problem with Padovan numbers and powers of two, Rev. Colombiana Mat. 53 (2019), no. 1, 1-14.
6 C. A. Gomez Ruiz and F. Luca, On the largest prime factor of the ratio of two generalized Fibonacci numbers, J. Number Theory 152 (2015), 182-203. https://doi.org/10.1016/j.jnt.2014.11.017   DOI
7 F. T. Howard and C. Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-242.
8 V. Iliopoulos, The plastic number and its generalized polynomial, Cogent Math. 2 (2015), Art. ID 1023123, 6 pp. https://doi.org/10.1080/23311835.2015.1023123   DOI
9 F. Luca, Fibonacci and Lucas numbers with only one distinct digit, Portugal. Math. 57 (2000), no. 2, 243-254.
10 R. Padovan, Dom Hans van der Laan: Modern Primitive, Architectura & Natura Press, Amsterdam, 1994.
11 I. Stewart, Tales of a neglected number, Sci. Amer. 274 (1996), 102-103.
12 D. Marques, On k-generalized Fibonacci numbers with only one distinct digit, Util. Math. 98 (2015), 23-31.
13 M. Mignotte, Sur les conjugues des nombres de Pisot, C. R. Acad. Sci. Paris Ser. I Math. 298 (1984), no. 2, 21.
14 M. Ddamulira, On the x-coordinates of Pell equations that are sums of two Padovan numbers, Bol. Soc. Mat. Mex. (3) 27 (2021), no. 1, Paper No. 4, 23 pp. https://doi.org/10.1007/s40590-021-00312-8   DOI
15 N. J. A. Sloane, The on-line encyclopedia of integer sequences, Ann. Math. Inform. 41 (2013), 219-234.
16 J. J. Bravo and F. Luca, On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math. Debrecen 82 (2013), no. 3-4, 623-639.   DOI
17 J. J. Bravo and F. Luca, On the largest prime factor of the k-Fibonacci numbers, Int. J. Number Theory 9 (2013), no. 5, 1351-1366. https://doi.org/10.1142/S1793042113500309   DOI
18 M. Ddamulira, Repdigits as sums of three Padovan numbers, Bol. Soc. Mat. Mex. (3) 26 (2020), no. 2, 247-261. https://doi.org/10.1007/s40590-019-00269-9   DOI
19 M. Ddamulira, Padovan numbers that are concatenations of two distinct repdigits, Math. Slovaca 71 (2021), no. 2, 275-284. https://doi.org/10.1515/ms-2017-0467   DOI
20 A. Alahmadi, A. Altassan, F. Luca, and H. Shoaib, Products of k-Fibonacci numbers which are rep-digits, Publ. Math. Debrecen 97 (2020), no. 1-2, 101-115.   DOI
21 J. J. Bravo, C. A. Gomez, and F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes 17 (2016), no. 1, 85-100. https://doi.org/10.18514/MMN.2016.1505   DOI
22 B. M. M. de Weger, Padua and Pisa are exponentially far apart, Publ. Mat. 41 (1997), no. 2, 631-651. https://doi.org/10.5565/PUBLMAT_41297_23   DOI