Not really the same as either of those, after trying long division I noticed that

(x^(k) - x^(k-1)) * (x^5 + x^4 + x^3 + x^2 + x + 1) = x^(k+5) - x^(k-1)

carrying that 1 step further

(x^(k) - x^(k-1) + x^(k-6) - x^(k-7) )*(x^5 + x^4 + x^3 + x^2 + x +1)=

x^(k+5) - x^(k-7) {equation a}

for the solution we have to divide

x^(5*12) + x^(4*12) + x^(3*12) + x^(2*12) + x^(1*12) + x^(0*12)

by (x^5 + x^4 + x^3 + x^2 + x + 1)

so setting k to 55, we will first have on the top of the long division:

x^55 - x ^54 + x^49 - x^48

{equation a} says this will be equal to x^60 - x^48 = x^(5*12) - x(4*12)

so the remainder, on the bottom of the long division, at this point, will be:

2x^(4*12) + x^(3*12) + x^(2*12) + x^(1*12) + x^(0*12)

by extension of the above we can see the remainder will follow the sequence:

1x^(5+12) + x^(4*12) + x^(3*12) + x^(2*12) + x^(1*12)

2x^(4*12) + x^(3*12) + x^(2*12) + x^(1*12)

3x^(3*12) + x^(2*12) + x^(1*12)

4x^(2*12) + x^(1*12)

5x^(1*12)

6

so we will stop the long division with the remainder 6x^(0*12), i.e. 6

the top of the long division will be the long looking polynomial:

+0

+1(x^55 - x^54 + x^49 - x^48)

+2(x^43 - x^42 + x ^37 - x^36)

+3(x^31 - x^30 + x^25 - x^24)

+4(x^19 - x^18 + x^13 - x^12)

+5(x^7 - x^6 + x - 1)