On Mon, 02 May 2011 10:53:52 +0100, Hans Georg Schaathun wrote:
On 02 May 2011 08:56:57 GMT, Steven D'Aprano
wrote:
: I see your smiley, but there are a number of similar series as
Fibonacci, : with the same recurrence but different starting values, or
similar but : slightly different recurrences. E.g. Lucas, primefree,
Pell, Padovan and : Perrin numbers.
Well, Fibonacci isn't one unique sequence. Any sequence satisfying f(n)
= f(n-1) + f(n-2) is /a/ Fibonacci sequence. Regardless of starting
values. At least according to some authors.
According to the references I have, there is one and only one Fibonacci
sequence, the usual one invented by Fibonacci to talk about rabbits.
(Actually, we should talk about Fibonacci *numbers*, rather than
sequence.)
Wolfram Mathworld is typical, defining *the* Fibonacci numbers as those
with starting conditions f(1)=1, f(2)=1 (or f(0)=0 if you prefer).
http://mathworld.wolfram.com/FibonacciNumber.htmlThe Collins Dictionary of Mathematics agrees with Mathworld.
The Lucas numbers (related to, but not the same as, the Lucas sequence)
obey the same recurrence as the Fibonacci numbers, but with a different
set of starting values. So there is certainly precedent in the
mathematical literature for giving different names to the same recurrence
with different starting values.
In any case, whatever you call them, what I call R(n) is not one, as the
recurrence is different:
R(n) = R(n-1) + R(n-2) + 1
Penguin, Dict. of Mathematics prefer 1,1,2,3,5 but they also suggest
0,1,1,2,3, ...
This depends on whether you start counting from n=0 or n=1.
Even the sequence you quote, from Anderson:
1,2,3,5...
is just the usual Fibonacci numbers starting at n=2 instead of 1 or 0.
(No matter how confusing something is, there's always at least one person
who will try to make it *more* confusing.)
In short, don't assume that a person talking about Fibonacci numbers
assume the same base cases as you do.
As far as I'm concerned, there are only two definitions of Fibonacci
numbers that have widespread agreement in the mathematical community:
0,1,1,2,3 ... (starting from n=0)
1,1,2,3,5 ... (starting from n=1)
Any other definition is rather, shall we say, idiosyncratic.
--
Steven