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What would be a closed-form formula that would determine the ith value of the sequence

1, 3, 11, 43, 171...

where each value is one minus the product of the previous value and 4?


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up vote 5 down vote accepted

The following is a simple way of solving the problem. The recurrence has the shape $$y_{n+1}=4y_n-1$$

It would be nice if the recurrence looked like $$u_{n+1}=4u_n$$

Then life would be easy. Unfortunately, we have that pesky $-1$ that spoils things!

So let's try for the next best thing. Could we transform our recurrence to something of the shape $$y_{n+1}-a=4(y_n-a)$$ where $a$ is a constant? Do a bit of arithmetic. We get $y_{n+1}=4y_n-3a$. If we choose $a=1/3$, we can get our recurrence into the desired shape.

Thus we can rewrite our recurrence as

$$y_{n+1} -\frac{1}{3}=4\left(y_n -\frac{1}{3}\right)$$

Temporarily, let $$u_n=y_n-\frac{1}{3}$$ for all $n$. Then $$u_{n+1}=4u_n$$

Now we have to decide whether we start our indices at $0$ or at $1$. Like many mathematicians, I prefer to start at $0$. (Starting at $1$ would not change things very much.)

We have $y_0=1$, so $u_0=1-1/3=2/3$. And because to get the "next" $u$ we multiply the previous $u$ by $4$, we have $$u_n=\left(\frac{2}{3}\right)4^n$$ But $y_n=u_n+1/3$. It follows that $$y_n=\left(\frac{2}{3}\right)4^n +\frac{1}{3}$$

Comment: Essentially the same method works for the recurrence $y_{n+1}=cy_n+d$ where $c$ and $d$ are constants, and the idea can be adapted to deal with many other situations.

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The general formula is:


This can be derived using the trick of converting a non-homogeneous difference equation, which in this case is $a_n=4a_{n-1}-1$, into a homogeneous one of a higher degree, which in this case is $a_{n+1}=5a_n-4a_{n-1}$. This has characteristic polynomial $x^2-5x+4=(x-4)(x-1)$, and therefore $$a_n=\alpha 4^n+\beta 1^n$$ for some $\alpha, \beta$. Solving the system $$1=a_0=\alpha+\beta$$ $$3=a_1=4\alpha+\beta$$ gives $\alpha=\frac{2}{3}$ and $\beta=\frac{1}{3}$.

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Thank you Zev! A really clean formulation. Is there a methodology you used? – Rio May 19 '11 at 9:14
@Rio: I've added an explanation of the method. This is the same process as the wikipedia article uses to solve the recurrence $a_n=a_{n-1}+1$. – Zev Chonoles May 19 '11 at 9:17

The OEIS gives the closed form formula of this sequence to be $$ \frac{2^{(2n+1)} + 1}{3}. $$

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Thank you for the quick response! – Rio May 19 '11 at 9:14

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