Interesting properties of number series

[gib65]

I'm currently taking an online course in mathematics. One of the subjects is about interesting properties that come out of certain number series (or number patterns).

 

Right off the bat, I'll spell out the purpose of this thread so you know where the OP is leading: I'm wondering if we can say that there's something "special" about these number series just because they bear certain interesting properties, or should we say (or prove) that there's always going to be interesting properties of any arbitrary number series no matter what it is?

 

As an example of a number series for which an interesting property emerges, take the following:

 

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

1 + 3 + 5 + 7 + 9 + 11 = 36

 

The number series in question is simply the odd positive integers. The interesting property that emerges is that the sum of the first n of them is always n^2.

 

I thought that was pretty interesting when I first heard it, but I couldn't figure out why that should be - until the lecturer gave a graphical explanation:

 

Take five squared. It can be represented graphically by a 5 x 5 square. Now do the following: count the number of square cells in the bottom left corner. Obviously, there's 1 of them. Write this down as the first number in the series. Now count the number of cells surounding this first cell. There's one above it, one diagonal to it (above and to the right), and one to its right. That's 3 cells. Write that down as the next number in the series. Do the same for the group of 3 cells. You will find that there are 5 cells surrounding it (we're not counting the 1 cell in the corner since that's already been counted). Write that down as the next number in the series. You should see the pattern here. Since after all the counting is done, you will have counted all the cells in the square, the sum of them should amount to the same as 5^2.

 

 

Here's another number series with a few interesting properties:

 

The Fibonacci numbers (defined as Fn = Fn-1 + Fn-2, starting with 0 and 1):

 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34...

 

Did you know that, starting from 0, every third Fibonacci number, and only every third, is even. Furthermore, starting with 0, every fourth, and only every fourth, is divisible by 3. Furthermore, starting with 0, every fifth, and only every fifth, is divisible, not by 4 this time, but 5. The pattern continues: every 6th is divisible by 8, every 7th by 13, every 8th by 21, etc. The pattern turns out to be:

 

Fn is divisible by Fm if and only if n is divisible by m (where F0 = 0).

 

An explanation for the even-ness of every third Fibonacci number was offered (which obviously by now should be interpreted as "divisible by 2"):

 

It's based on a couple simple rules of arithmetic:

 

1) an odd number added to an even number equals an odd number.

 

2) Two odd numbers added together equals an even number.

 

Since the Fibonacci sequence is defined by the addition of the last two preceding numbers, the above two rules can determine the parity of the next one. So starting with the first two Fibonacci numbers, the third's parity can be determined, and so on after that:

 

0 + 1 = 1 (even + odd = odd)

1 + 1 = 2 (odd + odd = even)

1 + 2 = 3 (odd + even = odd)

2 + 3 = 5 (even + odd = odd)

3 + 5 = 8 (odd + odd = even)

 

And so we see that the pattern repeats itself. And that is why every third is even.

 

I'm willing to bet that the divisibility of Fn by Fm can be explained by similar, or more general, rules of arithmetic.

 

Another interesting pattern that falls out of the Fibonacci series is as follows:

 

If GCD(m,n) denotes the greatest common denominator of m and n, then F(GCD(m,n)) = Fm, Fn. This is to be read that the greatest common denominator of the two Fibonacci numbers Fm and Fn is also a Fibonacci number - namely the GCD(m,n)th Fibonacci number.

 

In the course, the lecturer gives this example:

 

GCD(70, 90) = 10

 

Therefore, GCD(F70, F90) = F10.

 

Unfortunately, he didn't offer an explanation for this one.

 

I find all the above fascinating. It gives one (or me at least) the sense that there's something "special" about the Fibonacci sequence, and to a lesser extent, the odd integers - as though any other arbitrary number sequence wouldn't have nearly as many or interesting properties. But I wonder if this is true. Were I to invent a completely arbitrary number sequence, would I find, if I studied it deeply enough, that just as many and interesting properties emerge from it as well?

 

For example, let's say I defined the Gib numbers as follows:

 

1, 2, 2, 4, 8, 32, 256, 8192,...

 

That is, starting with 1 and 2, the next number is the product of the previous two:

 

1 x 2 = 2

2 x 2 = 4

2 x 4 = 8

4 x 8 = 32

8 x 32 = 256

32 x 256 = 8192

 

Right off the bat, I notice this series can be written:

 

2^F0, 2^F1, 2^F2, 2^F3, 2^F4, 2^F5, 2^F6, 2^F7,...

 

That is to say, the nth number in the Gib series is 2 raised to the nth Fibonacci number (where n begins at 0).

 

Ok, anything else? I'm not sure. Let me know if you find something.

 

What if I came up with something I bit more complex? Say:

 

1, 2, 6, 24, 120, 720,...

 

which is to say, starting with 1, the nth number is the (n-1)th times n (where n begins at 1).

 

Will there be just as many "neat" things about this number series as there are in the Fibonacci one or the odd number series? Will there be just as many in any arbitrarily constructed series?

Edited January 22, 2010 by gib65

[the tree]

"neat" isn't exactly a well defined term.

 

Some sequences can be defined as a function of the preceding terms (a recurrence relation) or as a function of it's position in the series.

 

For instance you might have a recurrence relation [imath]a_n = r \cdot a_{n-1}[/imath]. Each term would be something like: [math]a_0, r a_0, r^2 a_0, r^3 a_0, r^4 a_0, \dots[/math] and you could define the same function more simply as [imath]a(n) = r^n a_0[/imath]. Generally speaking, there's always going to be more than one way to write a given sequence.

 

Now, about your terminology. A series and a sequence aren't the same thing. (although at a low level it doesn't matter too much). A series is a sum of terms in a sequence (so you could say a series is a special type of sequence - but it has to have a sequence that it's based on).

 

So to take your first example:

 

[imath]a_n = 2n - 1 = 1,3,5,\dots[/imath] - a sequence of odd numbers.

 

[imath]S_N (a_n) = a_1 + a_2 + a_3 + \dots + a_N[/imath] - a series based on that sequence.

 

[imath]S_N (a_n) = N^2 = 1,4,9 \dots[/imath] - that series written as a sequence.

 

If you wanted to write a proper proof for that - to see how it works algebraically, then you'd have to know that:[math]S_N(n) = \frac{1}{2}(N^2 + N)[/math] which is a curious result but it comes with many applications.