Squares & Oblong numbers| 1*2 – 2*3 + 3*4 – 4*5 + … + (-1)^(n-1) n(n+1)

An Oblong number is a number of the form   $n \,(n + 1)$

Let   $A_n \; = \; (1\times 2) \; - \; (2\times 3) \; + \; (3\times 4) \; - \; (4\times 5) \; + \; ... \; + \; (-1)^{n-1} \; n \,(n+1)$

Determine the values of   $n$   for which

(1)   $2 \; \times \; \; A_n$   is a square number

(2)   $1/2 \; \times \; \; |A_n|$   is an Oblong number