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My favorite integer sequence
#36
Some variants :

In general f(n) = f(n-1) + f(n/a1) + f(n/a2) + ...
or f(n) = f(n-1) + f(n/a1) - f(n/a2) + f(n/a3) - ...

where 2<a1<a2<a3<... and all devisions are rounded to the smallest integer (floor). Also f(0) = 0 and f(1) = 1.

In particular

f(n) = f(n-1) + f(n/3) + f(n/5) + f(n/7) + f(n/9) + ...

and

g(n) = g(n-1) + g(n/3) - g(n/5) + g(n/7) - g(n/9) + ...

The growth rates of these 2 are fascinating.
Clearly this has number theoretic value.


also the sum 1/a1 + 1/a2 + ... is fascinating when it converges.

For instance

A(n) = A(n-1) + A(n/3)

B(n) = B(n-1) + B(n/4) + B(n/12)

Notice 1/3 = 1/4 + 1/12.

Does this imply that A(n) is close to B(n) ??

Many ideas and conjectures pop up.

regards

tommy1729
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Messages In This Thread
My favorite integer sequence - by tommy1729 - 07/27/2014, 08:44 PM
RE: My favorite integer sequence - by jaydfox - 07/31/2014, 01:51 AM
RE: My favorite integer sequence - by jaydfox - 08/01/2014, 01:53 AM
RE: My favorite integer sequence - by Gottfried - 08/01/2014, 03:25 PM
RE: My favorite integer sequence - by jaydfox - 08/01/2014, 05:04 PM
RE: My favorite integer sequence - by sheldonison - 08/01/2014, 11:36 PM
RE: My favorite integer sequence - by jaydfox - 08/01/2014, 11:57 PM
RE: My favorite integer sequence - by jaydfox - 08/05/2014, 04:49 PM
RE: My favorite integer sequence - by jaydfox - 08/05/2014, 05:57 PM
RE: My favorite integer sequence - by Gottfried - 08/06/2014, 04:38 PM
RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:19 AM
RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:34 AM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 09:42 AM
RE: My favorite integer sequence - by Gottfried - 08/09/2014, 02:27 PM
RE: My favorite integer sequence - by tommy1729 - 09/09/2014, 12:55 AM
RE: My favorite integer sequence - by jaydfox - 08/08/2014, 12:55 AM
RE: My favorite integer sequence - by Gottfried - 08/08/2014, 02:27 AM
RE: My favorite integer sequence - by jaydfox - 09/09/2014, 07:43 PM
RE: My favorite integer sequence - by jaydfox - 09/09/2014, 09:45 PM
RE: My favorite integer sequence - by jaydfox - 08/02/2014, 12:08 AM
RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:38 PM
RE: My favorite integer sequence - by sheldonison - 08/04/2014, 11:49 PM
RE: My favorite integer sequence - by jaydfox - 09/16/2014, 05:32 AM
RE: My favorite integer sequence - by Gottfried - 09/17/2014, 07:39 PM
RE: My favorite integer sequence - by jaydfox - 10/02/2014, 10:53 PM
RE: My favorite integer sequence - by Gottfried - 08/03/2014, 03:32 PM
RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:44 PM
RE: My favorite integer sequence - by sheldonison - 08/02/2014, 05:48 AM
RE: My favorite integer sequence - by tommy1729 - 09/10/2014, 08:57 PM
RE: My favorite integer sequence - by Gottfried - 08/02/2014, 07:43 PM
RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:29 PM
RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:36 PM
RE: My favorite integer sequence - by tommy1729 - 08/05/2014, 11:16 PM
RE: My favorite integer sequence - by tommy1729 - 08/08/2014, 11:02 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 07:02 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:51 PM
RE: My favorite integer sequence - by sheldonison - 08/11/2014, 04:51 PM
RE: My favorite integer sequence - by jaydfox - 08/11/2014, 05:19 PM
RE: My favorite integer sequence - by jaydfox - 08/19/2014, 01:36 AM
RE: My favorite integer sequence - by jaydfox - 08/19/2014, 02:05 AM
RE: My favorite integer sequence - by jaydfox - 08/19/2014, 05:31 PM
RE: My favorite integer sequence - by sheldonison - 08/19/2014, 07:56 PM
RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:42 AM
RE: My favorite integer sequence - by sheldonison - 08/20/2014, 02:11 PM
RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:57 PM
RE: My favorite integer sequence - by jaydfox - 08/21/2014, 01:15 AM
RE: My favorite integer sequence - by jaydfox - 08/21/2014, 05:25 AM
RE: My favorite integer sequence - by jaydfox - 08/22/2014, 05:39 PM
RE: My favorite integer sequence - by jaydfox - 09/11/2014, 01:33 AM
RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 09:16 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:19 PM
RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 10:52 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 11:46 PM
RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 11:10 PM
RE: My favorite integer sequence - by jaydfox - 08/10/2014, 12:30 AM
RE: My favorite integer sequence - by tommy1729 - 08/11/2014, 12:17 PM
RE: My favorite integer sequence - by Gottfried - 08/22/2014, 12:30 AM
Amazing variant - by tommy1729 - 08/26/2014, 08:57 PM
RE: My favorite integer sequence - by tommy1729 - 09/01/2014, 10:37 PM
RE: My favorite integer sequence - by tommy1729 - 10/02/2014, 11:24 PM
RE: My favorite integer sequence - by jaydfox - 10/02/2014, 11:29 PM
RE: My favorite integer sequence - by tommy1729 - 02/10/2015, 12:15 AM
RE: My favorite integer sequence - by tommy1729 - 02/15/2015, 05:19 PM
RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 08:22 AM
RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 09:10 PM
RE: My favorite integer sequence - by tommy1729 - 03/13/2016, 12:31 AM

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