The Square Numbers
The square numbers are: 1, 4, 9, 16, 25, 36, ...

The pattern is made clear by writing down what we have to add to each number to get the next: 

1   4   9   16   25    36 ...
  3   5   7    9    11    13 ...
Let's store the square number in S and the bit we add on in A - it will have to be increased by TWO each time.
  1. Load S with ONE
  2. Load A with THREE
  3. Output S
  4. Add A to S
  5. Increment A twice
  6. Go to step 3 

 LDA #1
 STA S
 LDA #3
 STA A
LOOP: OUT S
 LDA S
 ADD A
 STA S
 INC A
 INC A
 JMP LOOP

LDA #1
STA S
LDA #3
STA A
LOOP: OUT S
LDA S
ADD A
STA S
INC A
INC A
JMP LOOP

Assembler