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On Triangular Lucas Numbers

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Abstract

In the paper [3], we have proved that the only triangular numbers (i.e., the positive integers of the form \( \frac{1}{2}m \)(m+1)) in the Fibonacci sequence

$$ {u_n} + 2 = {u_{n + 1}} + {u_{{n^,}}}{u_0} = 0, {u_1} = 1 $$

are u ±1=u2=1, u4=3, u8=21 and u10=55. This verifies a conjecture of Vern Hoggatt [2]. In this paper we shall find all triangular numbers in the Lucas sequence

$$ {v_n} + 2 = {v_{n + 1}} + {v_{{n^,}}}{v_0} = 2, {v_1} = 1, $$

where n ranges over all integers.

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References

  1. Cohn, J. H. E. “On Square Fibonacci Numbers.” J. London Math. Soc. 39 (1964): pp. 537–541.

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  2. Guy, R. K. Unsolved Problems in Number Theory. New York: Springer-Verlag, 1981, p. 106.

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  3. Luo, Ming. “On Triangular Fibonacci Numbers.” The Fibonacci Quarterly, 27.2 (1989): pp. 98–108.

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© 1991 Springer Science+Business Media Dordrecht

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Luo, M. (1991). On Triangular Lucas Numbers. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3586-3_26

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  • DOI: https://doi.org/10.1007/978-94-011-3586-3_26

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5590-1

  • Online ISBN: 978-94-011-3586-3

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