Verifying a long polynomial equation in (the reciprocal of) the Golden Ratio I'm trying to show that the following equation holds true:

$$4\sigma^{12}+11\sigma^{11}+11\sigma^{10}+9\sigma^9+7\sigma^8+5\sigma^7+3\sigma^6+\sigma^5+\sigma^4+\sigma^3+\sigma^2+\sigma = 1 + 2\sigma$$
where $\sigma$ is the reciprocal of the golden ratio; that is, $\sigma := \frac12(\sqrt{5} - 1)$.

There must be a good way to show this. However, so far all my attempts have failed.
This problem has a real world application. If you are interested in background take a look at A Fresh Look at Peg Solitaire[1] and in particular at Figure 9.

[1] G. I. Bell [2007], A fresh look at peg solitaire, Math. Mag. 80(1), 16–28, MR2286485

As $\sigma$ verifies the identity $\sigma^2=-\sigma+1$, you can perform the following reductions (the powers of $\sigma$ have been omitted for conciseness):
$$4+11+11+9+7+5+3+1+1+1+1-1-1,\\
7+15+9+7+5+3+1+1+1+1-1-1,\\
8+16+7+5+3+1+1+1+1-1-1,\\
8+15+5+3+1+1+1+1-1-1,\\
7+13+3+1+1+1+1-1-1,\\
6+10+1+1+1+1-1-1,\\
4+7+1+1+1-1-1,\\
3+5+1+1-1-1,\\
2+4+1-1-1,\\
2+3-1-1,\\
1+1-1,\\
0+0.
$$
