Received August 24, 2023. Published online May 27, 2024.
Abstract: For any positive integer $k\geq2$, let $(P_n^{(k)})_{n\geq2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots,0,1$ ($k$ terms) with the linear recurrence
$P_n^{(k)} = 2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)}\quad\text{for} n\geq2$.
Let $(N_n)_{n\geq0}$ be Narayana's sequence given by
$N_0=N_1=N_2=1\quad\text{and}\quad N_{n+3}=N_{n+2}+N_n$.
The purpose of this paper is to determine all $k$-Pell numbers which are sums of two Narayana's numbers. More precisely, we study the Diophantine equation
$P_p^{(k)}=N_n+N_m$ in nonnegative integers $k$, $p$, $n$ and $m$.
Keywords: Diophantine equation; Narayana's cows sequence; $k$-Pell number; linear form in logarithms; reduction method
References: [1] J.-P. Allouche, T. Johnson: Narayana's cows and delayed morphisms. 3rd Computer Music Conference JIM96. IRCAM, Paris (1996), 6 pages.
[2] A. Baker, H. Davenport: The equations $3x^2-2=y^2$ and $8x^2-7=z^2$. Q. J. Math., Oxf. II. Ser. 20 (1969), 129-137. DOI 10.1093/qmath/20.1.129 | MR 0248079 | Zbl 0177.06802
[3] K. Bhoi, P. K. Ray: Fermat numbers in Narayana's cows sequence. Integers 22 (2022), Article ID A16, 7 pages. MR 4369871 | Zbl 1493.11032
[4] J. J. Bravo, P. Das, S. Guzmán: Repdigits in Narayana's cows sequence and their consequences. J. Integer Seq. 23 (2020), Article ID 20.8.7, 15 pages. MR 4161574 | Zbl 1477.11031
[5] J. J. Bravo, J. L. Herrera: Repdigits in generalized Pell sequences. Arch. Math., Brno 56 (2020), 249-262. DOI 10.5817/AM2020-4-249 | MR 4173077 | Zbl 07285963
[6] J. J. Bravo, J. L. Herrera, F. Luca: On a generalization of the Pell sequence. Math. Bohem. 146 (2021), 199-213. DOI 10.21136/MB.2020.0098-19 | MR 4261368 | Zbl 1499.11049
[7] Y. Bugeaud, M. Mignotte, S. Siksek: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann. Math. (2) 163 (2006), 969-1018. DOI 10.4007/annals.2006.163.969 | MR 2215137 | Zbl 1113.11021
[8] A. Dujella, A. Pethő: A generalization of a theorem of Baker and Davenport. Q. J. Math., Oxf. II. Ser. 49 (1998), 291-306. MR 1645552 | Zbl 0911.11018
[9] M. N. Faye, S. E. Rihane, A. Togbé: On repdigits which are sum or differences of two $k$-Pell numbers. Math. Slovaca 73 (2023), 1409-1422. DOI 10.1515/ms-2023-0102 | MR 4678548 | Zbl 7791584
[10] S. Guzmán Sanchez, F. Luca: Linear combinations of factorials and $S$-units in a binary recurrence sequence. Ann. Math. Qué. 38 (2014), 169-188. DOI 10.1007/s40316-014-0025-z | MR 3283974 | Zbl 1361.11007
[11] E. Kiliç: On the usual Fibonacci and generalized order-$k$ Pell numbers. Ars Comb. 88 (2008), 33-45. MR 2426404 | Zbl 1224.11024
[12] B. V. Normenyo, S. E. Rihane, A. Togbé: Common terms of $k$-Pell numbers and Padovan or Perrin numbers. Arab. J. Math. 12 (2023), 219-232. DOI 10.1007/s40065-022-00407-8 | MR 4552849 | Zbl 1523.11035
[13] N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences. Available at https://oeis.org/ (2019).
[14] Z. Wu, H. Zhang: On the reciprocal sums of higher-order sequences. Adv. Difference Equ. 2013 (2013), Paper ID 189, 8 pages. DOI 10.1186/1687-1847-2013-189 | MR 3084191 | Zbl 1390.11042
Affiliations: Kouèssi Norbert Adédji (corresponding author), Mohamadou Bachabi, Institute of Mathematics and Physics, Univeristy of Abomey-Calavi, Abomey-Calavi, Benin, e-mail: adedjnorb1988@gmail.com, mohamadoubachabi96@gmail.com; Alain Togbé, Department of Mathematics and Statistics, Purdue University Northwest, 2200 169th Street, Hammond, IN 46323, USA, e-mail: atogbe@pnw.edu
