Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.7

Repdigits in Narayana's Cows Sequence and their Consequences

Jhon J. Bravo
Departamento de Matemáticas
Universidad del Cauca
Popayán, Calle 5 No. 4-70

Pranabesh Das
Department of Pure Mathematics
University of Waterloo
200 University Ave. W.
Waterloo, ON N2L 3G1

Sergio Guzmán
Facultad de Ciencias en Física y Matemáticas
Universidad Autónoma de Chiapas
Blvd. Belisario Domínguez km. 1081
Tuxtla, C.P. 29050


Narayana's cows sequence satisfies the third-order linear recurrence relation Nn = Nn−1 + Nn−3 for n ≥ 3 with initial conditions N0 = 0 and N1 = N2 = 1. In this paper, we study b-repdigits that are sums of two Narayana numbers. As an illustration, we explicitly determine these numbers for the bases 2 ≤ b ≤ 100. We also obtain results on the existence of Mersenne prime numbers, 10-repdigits, and numbers with distinct blocks of digits in the Narayana sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms, and a version of the Baker-Davenport reduction method in diophantine approximation.

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(Concerned with sequences A000032 A000045 A000129 A000930 A010785.)

Received March 20 2020; revised versions received April 14 2020; October 11 2020. Published in Journal of Integer Sequences, October 12 2020.

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