In this post, we prove the closed form of a nonlinear recurrence corresponding to the count of binary trees with nodes.
This is inspired by an article of similar name called Calculating Fibonacci Numbers, Quickly and Exactly. Consider the a modified fibonacci sequence given by the recurrence . We shall look at various algebraic and analytic properties of this sequence, find two algorithms that computes in time, and finally look at a motivating example where we need to compute this sequence quickly and exactly.
Suppose that we are given the alphabet ; a word of length : , is an ordered -tuple whose elements all came from . For example, a word of length in might be the tuple , which we will hereon denote as . For some natural number , how many words of length are there that contains exactly zeros?
Suppose that we’re given the function , find the ordinary generating function associated with it in the form of . Furthermore, find/compute .
Suppose you have integers a,b that are relatively prime to m such that
Let , where k is an integer. Prove that for any positive integer n the number
is divisible by .
Define the sequence recursively by and
Find an explicit formula for in terms of n.
Let be defined by the recurrence relation , with . Show that the expression depends only on b and , but not on a.
Find the general term of the sequence given by , and