On Triangular Lucas Numbers

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.