req proof regarding A196

[franktaw at netscape.net]

The original description of this problem led me to look at the sequence
starting with n, the next term after m is m - floor(sqrt(m)), and take
the sum of the m's.  E.g., n=6, sequence 6,4,2,1, a(6) = 13.  This is
A076644.

What I noticed is that the first differences of A076644 form a fractal
sequence - which I have added as A122196.  Note that this, and
the related (also fractal) sequence A122197, are closely related to
the squares, and to A033638.

(There are a couple of typos in the references in these two
sequences; I just now sent Neil a correction.)

I haven't, at least so far, tried to follow the original construction.  
But
I strong suspect that the similarity of the initial description, and 
the
common reference to A033638, mean that there is a connection.

Franklin T. Adams-Watters


-----Original Message-----
From: Harris.Mitchell at mgh.harvard.edu

Joseph Biberstine wrote: 
 
>I suppose it is illuminative that A33638 are numbers of the form n^2+1 
>or n^2+n+1 (credit Donald S. McDonald), though I'm not quite bright 
>enough to see how. 
 
The "+1" is a red herring. 
 
Consider just the quarter squares floor(n^2/4) (A002620) which is also 
floor(n/2)ceil(n/2). 
 
If even you get 
  f(2n) = n^2, 
if odd you get 
  f(2n+1) = n^2+n = n(n+1) 
 
So not only does it have numbers both of those two forms but they also 
interleave. 
 
The complicated description was the combinatorial construction by the 
original sequence author which led to the sequence (similar to the Ulam 
description). 
 
Mitch 
 


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