[seqfan] Re: "near-Fibonacci sequences" and interval [0,1]

[Vladimir Shevelev]

Neil, for x=1/phi, I get the following near-Fibonacci sequence:
1,2,4,7,12,20,33,53,85,137,222,360,582,...
Note that there are many interesting questions concerning with this construction. Here only two from them:
1.  What values does take a part of those terms F_x(n) for which we have F_x(n)=F_x(n-1)+F_x(n-2) for various x from [1/3,1]? So, for x=1/3 and 2/3, it is 1.
2. Let us call two real numbers x and y from [1/3,1] equivalent, if the corresponding near-Fibonacci sequences {F_x(n)} and {F_y(n)} differ only on a finite set of n. What is the cardinality of the set {y}which are equivalent to x=1/3?
etc.
 
Best regards,
 
Vladimir


----- Original Message -----
From: "N. J. A. Sloane" <njas at research.att.com>
Date: Sunday, July 4, 2010 23:28
Subject: [seqfan] Re: "near-Fibonacci sequences" and interval [0,1]
To: seqfan at list.seqfan.eu
Cc: njas at research.att.com

> Vladimir, that's a nice idea! I wonder what real number
> you get if you use the binary expansion of 1/phi (or phi).
> .1001111000110111011110011011100101...,
> as the controlling sequence?
> 
>  Best regards
>                           Neil
> 
> 
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 Shevelev Vladimir‎