# Related sequences not related in database + question

Gerald McGarvey Gerald.McGarvey at comcast.net
Mon Nov 7 01:39:42 CET 2005

```are all substrings of {floor(sin(n))} contained in {floor(cos(n))} and
conversely ?

An attempt at a rough proof...
The cos(n) values and sin(n) values are both dense in (-1,1).
Given a cos(n) there is an m such that sin(m) is arbitrarily close to cos(n),
and also such that the derivative is arbitrarily close to the derivative of
cos(n).
In particular, given a substring of {floor(cos(n)} there is an m such that
sin(m)
is close enough to cos(n) (and the derivatives also close) so that the
substring of {floor(cos(n)} equals the substring of {floor(sin(n)} starting
at m
and of the same length (since cos and sin curves have the same shape).

Regards,
Gerald

At 06:09 PM 11/6/2005, Tautócrona wrote:
>Hi:
>
>In hot sequences I've seen the integer part of tan(n), namely the seq A000503
>
>In the database there'are also the integer parts of sin and cos and the
>rounding of sin
>and cos, but they don't appear in "related sequences" for A000503.
>
>Their seq numbers are, respectively, A000493, A000480, A000494, A000484
>
>Can anyone put all these sequences as related?
>
>By the way, are all substrings of {floor(sin(n))} contained in
>{floor(cos(n))} and
>
>Regards. Jose Brox
>http://espanol.groups.yahoo.com/group/Telecomunicacion/
>ambroxius at terra.es