Triangular numbers in the associated Pell sequence and Diophantine equations \(x^2(x+1)^2=8y^2\pm 4\). (Q1860752)

Integers of the form of \({m(m+1)\over 2}\) are called triangular numbers. \textit{M. Luo} has found all these numbers in the Fibonacci sequence \(F_{n+2}= F_{n+1}+ F_n\), \(F_0= 0\), \(F_1= 1\) [Fibonacci Q. 27, 98--108 (1989; Zbl 0673.10007)] and in the Lucas sequence \(L_{n+2}= L_{n+1}+ L_n\), \(L_0= 2\), \(L_1= 1\) [Applications of Fibonacci numbers, Vol. 4, 231--240 (1991; Zbl 0749.11016)]. \textit{W. L. McDaniel} has found all these numbers in the Pell sequence \(P_{n+2}= 2P_{n+1}+ P_n\), \(P_0= 0\), \(P_1= 1\) [Fibonacci Q. 34, 105--107 (1996; Zbl 0860.11007)]. Using the same method as in the above papers, the authors of this paper show that the only triangular numbers in the associated Pell sequence \(Q_{n+2}= 2Q_{n+1}+ Q_n\), \(Q_0= Q_1= 1\) are \(Q_0= Q_1= 1\) and \(Q_{\pm 2}= 3\). As a corollary they also obtain all integer solutions of the Diophantine equations \(x^2(x+1)= 8y^2\pm 4\).