In this post, we prove the closed form of a nonlinear recurrence corresponding to the count of binary trees with nodes.

# Currently Browsing

## Math Problems

# Calculating Modified Fibonacci Numbers, Quickly and Exactly

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.

# Counting Words

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?

# Annoying Mclaurin Series

Suppose that we’re given the function , find the ordinary generating function associated with it in the form of . Furthermore, find/compute .

# Power Reduction in Congruences

Suppose you have integers a,b that are relatively prime to m such that

then

# Even Pascals

Let , where k is an integer. Prove that for any positive integer n the number

is divisible by .

# Sequences in Sequences

Define the sequence recursively by and

Find an explicit formula for in terms of n.

# More Linear Recurrences

Let be defined by the recurrence relation , with . Show that the expression depends only on b and , but not on a.

# Almost Linear

Find the general term of the sequence given by , and

# Polynomial Divisors

Let . Show that for any positive integers there exists unique numbers such that the polynomial is divisible by p(x)