We now have a pattern for squared values, but what about cubed ones? Basically it is $ x^3 $.
I wasn't able to find a precise pattern that's similar to my answer on the linked post, as the result contains:
1^3 = 1
7
2^3 = 8
19
3^3 = 27
37
4^3 = 64
61
5^3 = 125
91
6^3 = 216
127
7^3 = 343
169
8^3 = 512
217
9^3 = 729
271
10^3 = 1,000
And I could barely make an equation basing it.
Without the help of a calculator, can I get the cubed values via patterns?
Well, the difference between consecutive cubes is
$$x^3-(x-1)^3= x^3-x^3+3x^2-3x+1=3x(x-1)+1$$
Three times the product of the bases plus one.