# Fibonacci numbers

Thomas Baruchel baruchel at bluebottle.com
Wed Nov 23 18:09:44 CET 2005

```Hi, it looks like I posted an idea of sequence to Neil one year ago.
But my interest then went to other things, and the stuff is not very
present in my mind. Maybe someone will be more interested, and I will
be happy to let him be the author and submitter of the sequence since
there is a little to do before the definition is complete.

Here is the idea:
(Note that the sequence starts almost like the true Fibonacci one).

----- Forwarded message from "N. J. A. Sloane" -----

Thomas, you said:

Date: Wed, 08 Sep 2004 16:03:30 +0200
Subject: An idea of sequence (generalized Fibonacci)

I noticed that the generalized sequence of Fibonacci [0,k,k,2k,3k,...]
can be computed with the following formula:
a(n) = round( k/sqrt(5) * phi^n )
BUT ONLY starting from a given offset: n > ???

what is the simplest 'order' (value of k) of the Fibonacci sequence such
that
the formula above works only starting from a given index:

Let's call a(n) the smallest value of k for which the formula works
starting from index n.
with n >= 2, we get:

2,3,5,8,13,21,33,53,85,138,223,361,583,943,1525,2468,3993,6461,10453,16913,27365,44278,...
(copied by hand; hope there is no mistake)

a(0) is probably 0...

I don't know what to put for a(1) ???
Explanation:

sequences of order 0 and 1 both have the formula working for n>=0.
sequence of order 2 has its formula working for n>=2

Thus no sequence has its formula working only when n>=1 which is the
definition of the sequence.

Is there a better definition for this sequence ?
Am I wrong somewhere ?
Is the sequence worth putting it in the OEIS ?

Me:  sorry for the delay in replying!  Yes, this should be
in the OEIS - can you please submit it in tghe usual way?

Thanks!

Neil

----- End forwarded message -----

--
Thomas Baruchel