I don't really understand how to solve (with generating functions) for the recurrence relation of $$a_n = a_{n-1}+2(n-1)$$ with initial conditions of $a_1 = 2$ when $n \geq 2$

This is what I was thinking, \begin{align*} g(x) &= a_0x^0+a_1x^1+a_2x^2+...+a_rx^r+...\\ g(x) &= a_0 + a_1x^1 + \sum^{\infty}_{n=2}a_nx^n\\ g(x) &= 1 + 2x + \sum^{\infty}_{n=2}{(a_{n-1}+2(n-1))x^n}\\ g(x) &= 1 + 2x + \sum^{\infty}_{n=2}a_{n-1}x^n + \sum^{\infty}_{n=2}(2n-2)x^n\\ g(x) &= 1 + 2x + x\sum^{\infty}_{n=1}a_{n-1}x^{n-1} + \sum^{\infty}_{n=2}2n x^n - \sum^{\infty}_{n=2}2x^n\\ g(x) &= 1 + 2x + x\sum^{\infty}_{m=1}a_{m}x^{m} + \sum^{\infty}_{n=2}2 \binom{n}{1}x^n - \sum^{\infty}_{n=2}2x^n\\ g(x) &= 1 + 2x + x(g(x) - a_0) + \sum^{\infty}_{n=2}2 \binom{n}{1}x^n - \sum^{\infty}_{n=2}2x^n\\ \end{align*}

  • $\begingroup$ A linear recurrence relation and two initial conditions? That's strange, but easily fixable. $\endgroup$ – Jack D'Aurizio Dec 12 '18 at 3:09
  • $\begingroup$ Fixed it -- thanks! $\endgroup$ – Violet Jung Dec 12 '18 at 3:22

You certainly do not need generating functions for solving $a_n-a_{n-1}=2(n-1)$: it is enough to sum both sides on $n=1,2,\ldots,N$ to get that $a_n$ depends on $\sum_{k=1}^{n}2(k-1)$. The only issue in your case is that the given values of $a_0$ and $a_1$ do not agree with the given recurrence relation. So, let us assume $a_0=\color{red}{2}$ and $a_{n}=a_{n-1}+2(n-1)$ for any $n\geq 1$. By denoting as $f(x)$ the following generating function $$ f(x)=\sum_{n\geq 0} a_n x^n = 2+\sum_{n\geq 1}a_n x^n$$ we have $$ x\cdot f(x) = \sum_{n\geq 0} a_n x^{n+1} = \sum_{n\geq 1} a_{n-1} x^n $$ hence by the recurrence relation $$ (1-x) f(x) = 2+2\sum_{n\geq 1}(n-1)x^n = 2+\frac{2x^2}{(1-x)^2}$$ where the last identity follows from stars and bars in the form $$ \frac{1}{(1-x)^{m+1}}=\sum_{n\geq 0}\binom{n+m}{m}x^n.$$ Since we have $$ f(x) = \frac{2}{1-x}+\frac{2x^2}{(1-x)^3} $$ stars and bars also gives $$ a_n = [x^n]f(x) = 2-n+n^2. $$

  • $\begingroup$ I was solving the question to "Find and solve the recurrence relation for the number of regions created by n mutually intersecting circles on a piece of paper (no three circles have a common intersection point)." so I incorrectly "set" the initial values. sorry! Do you have any recommendations for what the initial conditions should be? $\endgroup$ – Violet Jung Dec 12 '18 at 3:15
  • $\begingroup$ @MathNewbie: you may just start with $a_1=2$, the linear recurrence relation and the generating function defined as a sum over $n\geq 1$. $\endgroup$ – Jack D'Aurizio Dec 12 '18 at 3:18
  • $\begingroup$ makes sense - thank you. $\endgroup$ – Violet Jung Dec 12 '18 at 3:21
  • $\begingroup$ $a_1=2$ makes sense: a circle (simple closed curve) divides the plane into two connected components (Jordan's theorem). What is debatable is to assign a value to $a_0$. $\endgroup$ – Jack D'Aurizio Dec 12 '18 at 3:29
  • $\begingroup$ Sorry, understand that now. Thanks for the clarification $\endgroup$ – Violet Jung Dec 12 '18 at 3:29

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