Question 73374
{{{x^2+x-k}}}
= {{{x^2+2x(1/2)+(1/2)^2-(1/2)^2-k}}}
= {{{(x+1/2)^2-(1/2)^2-k}}}
= {{{(x+1/2)^2-1/4-k}}}
= {{{(x+1/2)^2-(1/4+k)}}}


This can be factorised if {{{k+1/4}}} is a perfect square.
Thus, those positive values of k which render {{{k+1/4}}} a perfect square are the reqd. values of k.
So, k = 2, 6, 12, 20, 30, etc.


[Note that the difference between the consecutive terms form an A.P.: 4,6,8,10,.... ]