Consider the sequence given by the recursion a₀ = 1, a₁=1/3, a_n+1 = 2/3*a_n - a_n-1 (n ≥ 1).  Prove that there exists a positive integer n such that a_n > 0.9999.

I wrote a program, and I found that a₂₄₅ = 0.999969287.  Can anybody give an exact mathematical proof?

Proof, by induction:

True for n = 0 and n = 1, because cos 0=1 and cos θ=1/3.

2/3 cos n θ - cos (n-1) θ  
= 2/3 cos nθ - (cos nθ cos θ + sin nθ sin θ),  from the sum of cosine identity
= (2/3 - cos θ) cos nθ - sin nθ sin θ
= 1/3 cos nθ - sin nθ sin θ,  because cos θ = 1/3 
= cos nθ cos θ - sin nθ sin θ,  again because cos θ = 1/3 
= cos (nθ + θ),  from the sum of cosine identity 
= cos (n+1)θ,
so by induction a_n = cos nθ for all n.

If θ/π is rational, a_n is periodic, so there exists an n such that a_n=a₀=1;
And if θ/π is irrational, nθ is dense in the circle so a_n is dense in [-1,1].

This proof doesn't depend on θ/π being irrational.  It's not hard to show that θ/π is irrational, though.

2 cos nθ is a monic polynomial in 2 cos θ with integer coefficients (obvious if you think about it, since 2 cos θ = e^iθ + e^-iθ). Hence if 2 cos nθ is 2 for any n > 0, 2 cos θ has to satisfy a nontrivial monic polynomial equation over , i.e. 2 cos θ is an algebraic integer. 2/3 is not an algebraic integer (it's not an integer). So a_n is never equal to 1, and thus θ/π is irrational. (Corollary: The only rational multiples of π that have rational cosines are integer multiples of π/3.)

As for the interesting patterns found in the investigation, 

cos^-1 1/3 = θ = 1.23095941734077

51θ = 62.7789302843795, and 10(2π) = 62.8318530717959, so 51θ is just 0.08% less than 10 trips around the circle.
46θ = 56.6241331976756, and 9(2π) = 56.5486677646163, so 46θ is just 0.13% more than 9 trips around the circle.

By interspersing 54 groups of 46θ, (about 9 trips around the circle) with 77 groups of 51θ, (about 10 trips around the circle), we get a total of 6411θ, which is almost exactly (to within 0.000001%, or one part in 100 million) 1256 trips around the circle.

The ratio of the number of θ's to the number of trips around the circle, 6411/1256=5.1043, is the primary observed period of this sequence.

The weighted average of 54 groups of 46θ and 77 groups of 51θ (131 groups in all) needed to make all these trips around the circle is about 6411/131=48.9389, which is the second-order period that was observed in the investigation.