Recurrences on Aurélien's Math Notebook

The Drunkard's Walk

Wed, 17 Oct 2018 00:00:00 +0000

A drunkard is zigzagging home. At every steps forward (or backward) he is making, he also moves $1$ step to the left with probability $p$ and $1$ step to the right otherwise. He starts $i$ steps to the right of a river. What is the expected number of steps forward before he falls into the river?

Recurrences

Sat, 22 Apr 2017 00:00:00 +0000

Theorem

Let $a \in \mathbb{N} $, $t < a \in \mathbb{N} $, and $b \in \mathbb{R}$. Defining $A_t$to be some real number and

$$ A_N = (1 - \frac{a}{N}) A_{N-1} + b, N > t \in \mathbb{N}, $$

then

$$ A_N = \frac{b}{1+a} (N+1), N \ge a \in \mathbb{N}. $$

Fibonacci numbers

Mon, 29 Jun 2015 00:00:00 +0000

The Fibonacci numbers are defined as $f_0 = 0,\ f_1 = 1$ and, for $i \ge 2,\ f_i = f_{i-1} + f_{i-2}$. Here is the beginning of the Fibonacci sequence:

$0, 1, 1, 2, 3, 5, 8, 13, 21, \ldots$

We generalize the definition above by changing the two initial values, for example with $f_0 = 4,\ f_1 = 6$ we obtain

$4, 6, 10, 16, 26, 42, \ldots$