All Questions

I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...

Let $T_q(n, k)$ be an integer table such that $$T_q(n, k) = \begin{cases} 1 & \textrm{if } n = 0 \vee k = 0 \\ qT_q(n-1, n-1) + T_q(n, n-1) & \textrm{if } n = k > 0 \\ T_q(n, k-1) + T_q(n-1,...

Let $a(n)$ be A347205. It is enough for us to know that $$ a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^jk), \\ a(0) = 1 $$ Let $b(n)$ be A329369. It is enough for us to know that $$ b(2^m(2k+1)) = \sum\...

Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\cdots,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...

For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$. For instance: $$ 100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...

Given a starting Penrose tile, I need to build a "spiraling" tessellation around it. The following picture illustrates the request: In this example, the starting tile is a "thin rhombus" (the pink ...