Calculate the Viswanath's constant (Random Fibonacci sequences)

Question

How Can I calculate first few digits of Viswanath's constant?

Viswanath’s constant ≈ 1.1319882487943 is a real number whose nth power approximates the absolute value of the nth term of some random Fibonacci sequences.

a[0] = 0;
a[1] = 1;
a[n_] := a[n] = RandomChoice[{-1, 1}]*a[n - 1] + RandomChoice[{-1, 1}]*a[n - 2]

I have MMA 12.1 with high-end PC but there is no answer even for $n=50000$

by definition Viswanath’s constant is a $n$ th root of $a[n]$

Block[{$RecursionLimit = Infinity}, N[a[50000]^(1/50000), 10]]

Answer 1

I think the problem arises because of the depth of recursion. If you evaluate smaller values first, the depth of recursion is much less.

For example, using your definition of a, the following evaluates in under a second

Block[{n = 100 i}, Table[a[n], {i, 1, 500}]];

memoizing the value for a[50000]

Answer 2

Maybe even a better solution is not to memoize the values at all:

Timing[a0 = 0; a1 = 1; n = 10^5;
 Do[{a0, a1} = {a1, a1 + RandomChoice[{-1, 1}]*a0}, {n - 1}];
 N[Abs[a1]^(1/n), 10]]
(*Out: {0.103201, 1.134360272}*)

But the convergence is slow.