looking for the pattern in these sequences: -2,0,1,4,4,8,_ , _ , _, _, and so forth

Answer 1

One way is to interlock sequences:

alternately
add 2,3,4,5,...
add 1,0,-1,-2,...

-2+2 = 0
0+1 = 1
1+3 = 4
4+0 = 4
4+4 = 8
8+(-1) = 7
7+5 = 12
12+(-2) = 10
...

Depending on your creativity, you can come up with arbitrarily bizarre rules, especially when there's no obvious simple pattern.