The Cube Numbers
The cube numbers are:
1x1x1, 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, ...
i.e. 1, 8, 27, 64, 125, 216, ...
You may have used multiplication and that's fine.
I used a similar method to the squares
because I noticed a pattern to the differences:
1   8    27    64    125    216 ...
  7   19    37    61     91 ...
    12   18    24    30 ...
       6     6     6
so this is what I ended up with:

 LDA #1
 STA N1
 LDA #7
 STA N2
 LDA #12
 STA N3
LOOP: OUT N1
 LDA N1
 ADD N2
 STA N1
 LDA N2
 ADD N3
 STA N2
 LDA N3
 ADD #6
 STA N3
 JMP LOOP 

LDA #1
STA N1
LDA #7
STA N2
LDA #12
STA N3
LOOP: OUT N1
LDA N1
ADD N2
STA N1
LDA N2
ADD N3
STA N2
LDA N3
ADD #6
STA N3
JMP LOOP

Assembler