Finding a Function to Generate a Particular Output

Date: 09/21/2004 at 14:26:35
From: Matt
Subject: creating a function with output 1,1,0,0,1,1,0,0

I am trying to figure out a formula that I could use to create a 
function such that

n    = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ...
f(n) = 0, 0, 1, 1, 0, 0, 1, 1, 0, 0  ...

I can't figure out how to create the repetition after two successive 
integers.  I know I can use (-1)^n  in order to create 0,1,0,1 ....
and I've tried using different takes on this idea... but I can't find 
anything.

Date: 09/21/2004 at 14:34:53
From: Doctor Vogler
Subject: Re: creating a function with output 1,1,0,0,1,1,0,0

Hi Matt,

Thanks for writing to Dr. Math.  It seems to me that you know what the
function is but just want a clever way to write it.  A mathematician
would usually write it as something like this:

           0  if n = 1 or 2 mod 4
  f(n) = {
           1  if n = 0 or 3 mod 4

If you are unfamiliar with modular arithmetic, it only means that the
remainder when you divide n by 4 is 0, 1, 2, or 3.  So you could also
write this as:

           0  if n = 4k + 1 or n = 4k + 2 for some integer k
  f(n) = {
           1  if n = 4k or n = 4k + 3 for some integer k

On the other hand, if you really want to do the (-1)^n thing, then you
can write f(n) as

  f(n) = (1/2)(1 + (-1)^[n(n+1)/2]).

That looks confusing.  Let's write it as

                    n(n+1)/2
          1  +  (-1)
  f(n) = --------------------
                 2

Then if n is 0 or 3 mod 4, then either n or n+1 is divisible by 4, so
that the exponent will be even, and you will get (1+1)/2.  If n is 1
or 2 mod 4, then one of n and n+1 will be odd and the other will be
even but not a multiple of 4, so the exponent will be odd, and you
will get (1-1)/2.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/