-21, -9, 11, 39
it appears that the formula for this would be:
a[1] = -21
a[n] = a[n-1] + 12 + 8*(n-2)
we start with:
a[1] = -21
a[2] = -21 + 12 + 8*0 = -21 + 12 = -9
a[3] = -9 + 12 + 8*1 = -9 + 12 + 8 = -9 + 20 = 11
a[4] = 11 + 12 + 8*2 = 11 + 12 + 16 = 39
this would make:
a[5] = 39 + 12 + 8*3 = 39 + 12 + 24 = 75
if this formula is correct, then the next number in the sequence is equal to 75.
the only pattern i could find is this.
-21
-21 + 12 = -9
-9 + 20 = 11
11 + 28 = 39
if you look at the differences of 12, 20, and 28, you will see that the difference increases by 8 each time.
this means that the next number in the sequence would have to be 39 + 36, because 36 is equal to 28 + 8.
39 + 36 is equal to 75.
i derived the formula i used from this observation.