Related sequences not related in database + question

[franktaw at netscape.net]

floor(cos(0)) = 1, which does not occur in floor(sin(n)); but other than that, all finite substrings in either occur in the other.  Both sequences are of the form floor(sin(n+K)), and the only case where (n+K)/(2Pi) is an integer is for n=0.  So leaving aside n=0, the value of floor(sin(x)) in some neighborhood of n+K is constant.  These intervals can be translated back to the starting point of the sequence, and intersected to give a single interval.  Since multiples of 1 mod 2Pi are dense, we can find n'+K' which is in that interval, providing the starting point for the matching sequence.  For the case n=0 with the sin function, we are restricted to a half-open interval starting at zero, but otherwise the same argument works.  (And, as indicated, n=0 for the cosine does not work.)
 
The same kind of argument will establish that the sequences for rounded sin and cos contain all of each other's finite subsequences.
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
 
 
-----Original Message-----
From: Tautócrona <tautocrona at terra.es>
To: seqfan at ext.jussieu.fr
Sent: Mon, 7 Nov 2005 00:09:00 +0100
Subject: Related sequences not related in database + question


...
By the way, are all substrings of {floor(sin(n))} contained in {floor(cos(n))} 
and 
conversely ? Any results about this type of statements?

Regards. Jose Brox
http://espanol.groups.yahoo.com/group/Telecomunicacion/
ambroxius at terra.es
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