Suppose we have a sequence $a$ with a repeating pattern of length $k$. For simplicity, we can express it as: $$ a=\{c_0,c_1,...,c_{k-1},c_0,c_1,...\} $$ Where $a_n$ is the $n$th term, starting from $n=0$, can a generating function be found for such a sequence?

Dr Buffy is at 308,000 and counting!

As Pete L. Clark mentions in his excellent answer the statement "the primes are considered to be the 'building blocks' of the integers" is a soundbite - a shorthand way to refer to a very important fact about the natural numbers and by extension, of the integers. The additive structure of the i...

Take any scientific calculator and follow these steps: Ensure the calculator is in degrees mode. Enter some number of 5's (more than 4 is ideal). Take the reciprocal of step 2. Take the sine of step 3. You should find that the result is an approximation of $\pi$ (that becomes more accurate as m...

Consider the following exact sequence: $0\to \mathbb Z^2\to H_1(X,A)\to \mathbb Z^{k-1}\to 0$. How do I show that this splits? By definition of splitting, I should have $H_1(X,A)= \mathbb Z^2 \oplus \mathbb Z^{k-1}=\mathbb Z^{k+1}$. But I don't understand how to show that this is the case here. T...

Let $B$ be a standard Brownian motion. Define for all $n\geq 1$ the functions $B^{(n)}$ s.t. for all $t\in 2^{-n}\Bbb{N}_0$ $$B_t^{(n)}=B_t$$ and such that $B^{(n)}$ is linear on the intervals $\left[\frac{k}{2^n}, \frac{k+1}{2^n}\right]$ for all $k\geq 0$. I want to show that for a fixed $\epsi...

Numerical methods for approximating Pythagoras' constant $t =\sqrt 2$ by fractions. (This is an idea from my mentor while he was barely $13$ yo, as a response to a challenge). We all know Newton's method for finding $t$. It converges quadratically meaning like $o( C x^{2^n} )$ where $C$ is a cons...