Extensions of the D(∓k2)-triples {k2, k2 ± 1, 4k2 ± 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k2, k2 + 1, 4k2 + 1, d} is a D(−k2)-quadruple, then d = 1, and that if {k2 − 1, k2, 4k2 − 1, d} is a D(k2)-quadruple, then d = 8k2(2k2 − 1).