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THE SQUARE TERMS IN GENERALIZED LUCAS SEQUENCES
Author(s) -
Şi̇ar Zafer,
Keski̇n Refi̇k
Publication year - 2014
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579313000193
Subject(s) - mathematics , fibonacci number , lucas sequence , sequence (biology) , lucas number , zero (linguistics) , square (algebra) , fibonacci polynomials , combinatorics , mathematical analysis , geometry , linguistics , philosophy , biology , genetics , orthogonal polynomials , difference polynomials
Let P and Q be non‐zero integers. The generalized Fibonacci sequence { U n } and Lucas sequence { V n } are defined byU 0 = 0 ,U 1 = 1 andU n + 1 = P U n + Q U n ‐ 1for n ≥ 1 andV 0 = 2 , V 1 = P andV n + 1 = P V n + Q V n ‐ 1for n ≥ 1 , respectively. In this paper, we assume that Q = 1 . Firstly, we determine indices n such thatV n = k x 2when k | P and P is odd. Then, when P is odd, we show that there are no solutions of the equationV n = 3 □ for n > 2 . Moreover, we show that the equationV n = 6 □ has no solution when P is odd. Lastly, we consider the equationsV n = 3 V m □ andV n = 6 V m □ . It has been shown that the equationV n = 3 V m □ has a solution when n = 3 , m = 1 , and P is odd. It has also been shown that the equationV n = 6 V m □ has a solution only when n = 6 . We also solve the equationsV n = 3 □ andV n = 3 V m □ under some assumptions when P is even.