#x86, 41 bytes
Mostly straightforward implementation of the formula with input in ecx and output on the stack.
The interesting thing is that I used a cubing function, but since call label is 5 bytes, I store the label's address and use the 2 byte call reg. Also, since I'm pushing values in my function, I use a jmp instead of ret. It's very possible that being clever with a loop and the stack can avoid calling entirely.
I did not do any fancy tricks with cubing, like using (k+1)^3 = k^3 + 3k^2 + 3k + 1.
Edit: Fix byte count using not instead of neg/dec.
.section .text
.globl main
main:
mov $10, %ecx # n = 10
start:
lea (cube),%edi # save function pointer
call *%edi # output n^3
sub %ecx, %eax # n^3 - n
xor %edx, %edx
push $6
pop %ebx # const 6
idiv %ebx # k = (n^3 - n)/6
mov %eax, %ecx # save k
call *%edi # output k^3
push %eax # output k^3
not %ecx # -k-1
call *%edi # output (-k-1)^3
inc %ecx
inc %ecx # -k+1
call *%edi # output (-k+1)^3
ret
cube: # eax = ecx^3
pop %esi
mov %ecx, %eax
imul %ecx
imul %ecx
push %eax # output cube
jmp *%esi # ret
Objdump:
00000005 <start>:
5: 8d 3d 24 00 00 00 lea 0x24,%edi
b: ff d7 call *%edi
d: 29 c8 sub %ecx,%eax
f: 31 d2 xor %edx,%edx
11: 6a 06 push $0x6
13: 5b pop %ebx
14: f7 fb idiv %ebx
16: 89 c1 mov %eax,%ecx
18: ff d7 call *%edi
1a: 50 push %eax
1b: f7 d1 not %ecx
1d: ff d7 call *%edi
1f: 41 inc %ecx
20: 41 inc %ecx
21: ff d7 call *%edi
23: c3 ret
00000024 <cube>:
24: 5e pop %esi
25: 89 c8 mov %ecx,%eax
27: f7 e9 imul %ecx
29: f7 e9 imul %ecx
2b: 50 push %eax
2c: ff e6 jmp *%esi
Here is my testing version that does all the cubing at the end. After the values are pushed on the stack, the cube loop overwrites stack values. It's currently 42 bytes but there should be some improvements somewhere.
.section .text
.globl main
main:
mov $10, %ecx # n = 10
start:
push %ecx # output n
mov %ecx, %eax
imul %ecx
imul %ecx
sub %ecx, %eax # n^3 - n
xor %edx, %edx
push $6
pop %ecx # const 6
idiv %ecx # k = (n^3 - n)/6
push %eax # output k
push %eax # output k
not %eax # -k-1
push %eax # output -k-1
inc %eax
inc %eax # -k+1
push %eax # output -k+1
dec %ecx # count = 5
add $20, %esp
cube:
mov -4(%esp),%ebx # load num from stack
mov %ebx, %eax
imul %ebx
imul %ebx # cube
push %eax # output cube
loop cube # --count; while (count)
ret