How to speed up series generation?

Question

The problem requires to generate the n-th element of a sequence that is similar to Fibonacci sequence. However, it\'s a bit tricky because n is ver

Answer 1

Just thinking here but take a look at Duff's device for the count_fair_coins function as that will automatically unroll the loop to speed up that function.

Precomputing the sqrt's in the generating function seems like the easiest way to get any speed up. Which would reduce to just a single pow call and multiplication of constants. As well as precomuting the sqrt's another way to speed it up is to remove the divisions and use the inverse multiplication, although a very slight optimization it might help speed up when n is very large.

Answer 2

You can write the terms of the sequence in terms of matrix exponentials:

which can be quickly evaluated using exponentiation by squaring. This leads to an O(log n) solution which should solve the problem well within the time constraints.

Just for future reference, if you are required to do multiplication with large numbers (not applicable in this situation since the answer is taken modulo 1000000007), you should look into the Karatsuba algorithm. This gives you sub-quadratic time multiplication.