(In reply to computer solution
a. Your count of possible solutions to a*b solutions is correct, mine and Jer's are not , counting twice the part with negative numbers.
b. However since "arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant" the first two entries hardly qualify:
2016 - there are no successive members,
1007,1009 - no constant difference.
So the total number of arithmetic sequences is 34.