How do I evaluate ? Note that it's equivalent to solving the recurrence relation for .
I know how to prove it's between .8 and 1, but I can't figure out an exact answer.
here. That link gives the expansion of as . It's not stated in the link, but it looks very much as though the nonzero coefficients are all (with a pattern of two –1s followed by two +1s). If so then with an error of less than .
In fact, that pattern of pairs of ±1s looks extremely regular, with the k'th pair being the coefficients of and , where and . Again, that's just based on observations from the list of the first 200 coefficients in the Encyclopedia of Integer Sequences. I don't have any proof for it.
A reasonable alternative to the 'direct computation' of the 'infinite product' is the computation of its logarithm...
Using the standard Taylor expansion of we obtain...
Both the 'infinite product' and the 'infinite sum' are convergent for . The computation using the first three hundred terms of the 'product' and the first three hundred terms of the 'sum' leads us to the same result approximated in twelve digits...
Because of the formula
we have (only ones and zeros).
Note as well that . Therefore,
There may be mistakes, but you see the idea: has a decimal expansion with only 0,1,2's. And because the exponents in the first formula are quadratic (not linear), this expansion is not eventually periodic.
We conclude that is irrational, and thus so is .