NK ladder and snake

#1

In the example, Snakes were present at (n -1)th position and (n-4)th position, then you mentioned recursive formula will given as,
F(i) = 0 * F(i-1) + 1 * F(i -2) + 1 * F(i-3)
but if this the case then when we would be computing
F(i - 2) in the recursive process then its formula would become F(i - 2) = 0* F(i - 3) + 1 * F(i - 4) + 1 * F(i - 5)
here we should have made F(i - 4) to be zero because snake is present at that position but instead we are making F(i - 3) to be zero where there is no snake.
So, is this formula correct if so please explain it.

#2

We will have to change coefficients with recursive calls.
You are thinking right that if we keep (0,1,1) in every call, it will result to a wrong answer.
So we change the variables in every call.

#3

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#4

If we have to change the variable in every call then complexity will become k^3n instead of k^3logn this is even more than dp case where it is only n what is the purpose of using this