Find a recursive definition for this set of numbers, visual

Question

I have been having trouble trying to find a recursive definition for this following visual sequence of numbers. Thank you for your help/suggestions.

The numbers I've derived so far are:

\begin{align} F(1) & = 1 \\ F(2) & = 5 \\ F(3) & = 15 \\ F(4) & = 30 \\ F(5) & = 50 \end{align}

This means the differences between this are $4, 10, 15, 20$.

Answer 1

Let $a_n$ denote the number of dots where $n$ is the number of concentric Pentagons. We can arbitrarily set $a_0 = 1$ (but do be aware that this will not hold in the recursion)

The nth pentagon always has $5$ vertex dots, and $n-1$ edge dots (with 5 sides).

So you get $$a_n = a_{n-1} + 5(n-1) + 5 = a_{n-1} + 5n $$

We can also get an explicit form:

$$a_2 = a_1 + 5*2 = 5 + 5*2$$ $$a_3 = a_2 + 5*3 = 5 + 5*2 + 5*3$$ $$a_4 = 5(1 + 2 + 3 + 4) $$ ... $$a_n = 5 \sum_{k=1}^{n} k = \frac{5n(n+1)}{2}$$