Sequence

**novice** · July 9th 2010, 12:19 PM

Let $a_0, a_1, a_2,$ ... be a seqence of positive integers for which

$a_0=1,$
$a_{2n+1}=a_n$
$a_{2n+2}=a_n+a_{n+1}$

Prove that $a_n$ and $a_{n+1}$ are relatively prime for every nonnegative integer $n$ .

$a_0=1$
$n=0, \:a_1=1, \:\:a_2=2$
$n=1, \:a_3=1, \:\:a_4=3$
$n=2, \:a_5=2, \:\:a_6=3$
$n=3, \:a_7=1, \:\:a_8=4$
$n=4, \:a_9=3, \:a_{10}=5$
$n=5, \:a_{11}=2, \:a_{12}=5$
$n=6, \:a_{13}=3, \:a_{14}=4$
$n=7, \:a_{15}=1, \:a_{16}=5$
$n=8, \:a_{17}=4, \:a_{18}=7$
...
...
...
There doesn't seem to be a discernable pattern, but any how, I think these can be approached by Strong Induction.

The basis case is easy. $gcd(a_0,a_1)=gcd(1,1)=1$ is obvious.

The Inductive step: Assume $gcd(a_k,a_{k+1})=gcd(a_{2p+1}, a_{2p+2})=gcd(a_p,a_{p+1})=1$ . We show that $gcd(a_{k+1},a_{k+2})=1$ , $p \in \mathsb{z}^+ \cup {0}$

I can't see the recurrence relation for $a_k$ . Need help.

**novice** · July 9th 2010, 03:05 PM

I think I have found the general expression for $a_k$ .

$a_k=a_{k-1}+(-1)^k a_{k-2}$

**Ackbeet** · July 9th 2010, 03:56 PM

I'm a tad confused by the OP: isn't $a_{2n+2}=a_{2n+2}$ a tautology? What is the original problem, here?

**chiph588@** · July 9th 2010, 05:10 PM

Originally Posted by novice

$a_0=1$
$n=0, \:a_1=1, \:\:a_2=2$
$n=1, \:a_3=1, \:\:a_4=3$
$n=2, \:a_5=2, \:\:a_6=3$
$n=3, \:a_7=1, \:\:a_8=4$
$n=4, \:a_9=3, \:a_{10}=5$
$n=5, \:a_{11}=2, \:a_{12}=5$
$n=6, \:a_{13}=3, \:a_{14}=4$
$n=7, \:a_{15}=1, \:a_{16}=5$
$n=8, \:a_{17}=4, \:a_{18}=7$

This sequence is Stern's Diatomic Series.

Originally Posted by Ackbeet

I'm a tad confused by the OP: isn't $a_{2n+2}=a_{2n+2}$ a tautology? What is the original problem, here?

Stern's Diatomic Series is defined as $a_0=0,\;a_1=1,\;\begin{cases}a_{2n}=a_n,\; n\text{ is even}\\a_{2n+1}=a_n+a_{n+1},\; n\text{ is odd}\end{cases}$

Now for the proof, base case: $(a_0,a_1)=(0,1)=1$ .

Assume $(a_m,a_{m+1})=1$ for all $m<k$ .

Case 1: $k=2n$ which gives $(a_{2n},a_{2n+1})=(a_n,a_n+a_{n+1})$ .

By the IH, there exists $x,y\in\mathbb{Z}$ such that $xa_n+ya_{n+1}=1\implies (x-y)a_n+y(a_n+a_{n+1})=1\implies (a_n,a_n+a_{n+1})=1$

Therefore $(a_k,a_{k+1})=1$

Case 2: $k=2n+1$ which gives $(a_{2n+1},a_{2n+2})=(a_n+a_{n+1},a_{n+1})$ .

By the IH, there exists $x,y\in\mathbb{Z}$ such that $xa_n+ya_{n+1}=1\implies x(a_n+a_{n+1})+(y-x)a_{n+1}=1\implies (a_n+a_{n+1},a_{n+1})=1$

Therefore $(a_k,a_{k+1})=1$

So by strong induction the claim $(a_n,a_{n+1})=1$ for all $n\in\mathbb{N}$ holds.

**novice** · July 9th 2010, 06:35 PM

I apologize. It's $a_{2n+2}=a_n+a_{n+1}$

**novice** · July 9th 2010, 06:40 PM

You are quite amazing. You made it look so simple.

**chiph588@** · July 9th 2010, 06:55 PM

Originally Posted by novice

You are quite amazing. You made it look so simple.

You'll have to modify the proof a bit, as our definitions of the sequence slightly differ.

**chiph588@** · July 9th 2010, 07:00 PM

Originally Posted by novice

I think I have found the general expression for $a_k$ .

$a_k=a_{k-1}+(-1)^k a_{k-2}$

Fyi this is wrong as it says $a_6=5$ when actually $a_6=3$ .

**novice** · July 9th 2010, 08:43 PM

Originally Posted by chiph588@

Fyi this is wrong as it says $a_6=5$ when actually $a_6=3$ .

Yes, thank you. I plotted the results, and saw that the curves didn't meet. I felt embarrassed.
You didn't do it to me, but I did to myself quite often.

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