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Posts Tagged ‘ algebra ’

29 December

Counting Binary Trees (The Hard Way)

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

9 November
Posted in Article, Math Problems

Counting Words

Suppose that we are given the alphabet \Sigma = \Rec{0,1,2}; a word of length n: x_n \in \Sigma^5, is an ordered n-tuple whose elements all came from \Sigma. For example, a word of length 5 in \Sigma might be the tuple (0,0,2,1,0), which we will hereon denote as 00210. For some natural number k \le n, how many words of length n are there that contains exactly k zeros?

6 November

Annoying Mclaurin Series

Suppose that we’re given the function B(z) = \frac{1 + z}{1 - z - z^2}, find the ordinary generating function associated with it in the form of B(z) = \sum_{k=0}^\infty b_k k^z. Furthermore, find/compute \sigma_n = \sum_{k=0}^n b_k.

14 August
Posted in Article, Math Problems

Even Pascals

Let a=4k-1, where k is an integer. Prove that for any positive integer n the number

     $$ s_n = 1 - {n \choose 2}a + {n \choose 4}a^2 - {n \choose 6}a^3 + \cdots $$

is divisible by 2^{n-1}.

14 August
Posted in Article, Math Problems

Sequences in Sequences

Define the sequence (a_n)_n recursively by a_1 = 1 and

     $$ a_{n+1} = \frac{1 + 4a_n + \sqrt{1+24a_n}}{16} ,\hspace{4mm} \mbox{for $n \ge 1$.} $$

Find an explicit formula for a_n in terms of n.

14 August
Posted in Article, Math Problems

More Linear Recurrences

Let (x_n )_n = 0 be defined by the recurrence relation x_{n + 1} = ax_n + bx_{n - 1}, with x_0 = 0. Show that the expression x^2_n - x_{n - 1} x_{n + 1} depends only on b and x_1, but not on a.

13 August
Posted in Article, Math Problems

Almost Linear

Find the general term of the sequence given by x_0 = 3, x_1 = 4, and

    $$(n + 1 )(n + 2 )x_n = 4 (n + 1 )(n + 3 )x_{n - 1} - 4 (n + 2 )(n + 3 )x_{n - 2}$$

13 August
Posted in Article, Math Problems

Polynomial Divisors

Let p(x) = x^2 -3x + 2. Show that for any positive integers  n \ge 2 there exists unique numbers a_n, b_n such that the polynomial  x^n - a_n x - b_n is divisible by p(x)

13 August
Posted in Article, Math Problems

Analytical Fibonacci

We derive Binet’s equation for the nth Fibonacci number as

     $$ F_n = \frac{1}{\sqrt 5}\left( \left(\frac{1+\sqrt 5}{2}\right)^n -  \left(\frac{1-\sqrt 5}{2}\right)^n  \right) $$

13 August
Posted in Article, Math Problems

Linear Recursive Sequence

We derive a general technique for solving full rank linear recursive sequences. Formally, a kth linear recursive sequence is defined as

     $$ x_n = a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3} + \stackrel{k}{\ldots} + a_k x_{n-k} $$