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

Work with $x$ instead of $\sigma$. You have $$x^2=-x+1$$ so $$x^3=x-x^2=+2x-1$$ $$x^4=x^2-x^3=-3x+2$$ and you will find (without surprise) that the Fibonacci numbers appear, so you can write down $$x^5=+5x-3, x^6=-8x+5 \dots$$ and this may be an efficient way of reducing to a linear expression.
