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  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.
  
  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/
  ```
  Associated Topics:
  High School Sequences, Series
  
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