Fascination of Sequences (was: Re: breaking 50K)

[Antti Karttunen]

Michael Somos wrote:
> 

> Anyone care to comment about the fascination of integer sequences? What
> is it about the Fibonacci sequence, or Catalan numbers that seems so
> attractive to some people? I know that early in my life I became very
> interested in numbers and especially sequences of numbers.


 I think the foremost cause is this:
 The most of the sequences in the Encyclopedia are infinite.
 Infinity is always fascinating.
 Then we have to separate two cases:
 
 Sequences which have no obvious formula, and which must
 be produced by some kind of a filtering/checking process.
 I.e. primes, (also of special format like Fermat or
 Mersenne primes), Perfect Numbers, and such.
 Here we have the enthusiasm of a treasure hunter.
 Where I can find yet another term? *Is* there yet another
 term? Often coupled with this we have the desire
 to seek for primes from all the unlikely places.
 (Because of our Platonic Ideals, we think they must exist
  somewhere out there...)
  
 Sequences with formula, whether direct or recursive.
 Here we have infinitely many terms, yet they are all
 bound to each other with an underlying, often very
 simple law. Like the notes of the partiture, also
 bound by the underlying theme.
 And maybe we can find a counterpoint or transposition,
 no, I mean transformation that gives yet another elegant
 sequence.
 
 And of course the fascination of the Fibonacci numbers
 themselves stems partly from their incredible ability
 to stick out their rabbit-heads from so many fields
 of mathematics. Continued fractions? Yes, consider
 the simplest one, [1,1,1,1,...]. Linear algebra?
 Just start taking powers of matrix
 (1 1)
 (1 0)
 just to name a few.


> Shalom, Michael

Yours,

Antti

E-mail: Antti.Karttunen at iki.fi -> karttu at megabaud.fi