[seqfan] Re: Fwd: Iterates of n - A002828(n) ?

L. Edson Jeffery lejeffery2 at gmail.com
Mon Aug 19 21:02:17 CEST 2013


AK>...will the trajectory visit every square-1 (A005563) < k before it
reaches zero?

Is it sufficient to prove, for any n, that max(square-1 < n) is in the
trajectory?

AK>Which would imply that there would be only one infinite sequence a_n
such that a(n-1) = a(n) - the least number of squares that add up to a(n).

This should be essentially the first column in the first triangle below.

AK>Then, would there be any regularity in the derived sequence counting the
number of iterations needed to hop from A005563(n+1) to A005563(n)?

According to the second triangle below, it appears that your hopping
sequence should be {1,1,2,2,4,4,5,...}, unless I misunderstood or
miscalculated. Not sure exactly what you mean by "regularity."

Nice ideas, Antti.

 n     Seq of trajectories for 0 <= n <= 48
---   -----------------------------------------
 0     0
 1     0
 2     0
 3     0
 4     3,0
 5     3,0
 6     3,0
 7     3,0
 8     6,3,0
 9     8,6,3,0
10     8,6,3,0
11     8,6,3,0
12     9,8,6,3,0
13     11,8,6,3,0
14     11,8,6,3,0
15     11,8,6,3,0
16     15,11,8,6,3,0
17     15,11,8,6,3,0
18     16,15,11,8,6,3,0
19     16,15,11,8,6,3,0
20     18,16,15,11,8,6,3,0
21     18,16,15,11,8,6,3,0
22     19,16,15,11,8,6,3,0
23     19,16,15,11,8,6,3,0
24     21,18,16,15,11,8,6,3,0
25     24,21,18,16,15,11,8,6,3,0
26     24,21,18,16,15,11,8,6,3,0
27     24,21,18,16,15,11,8,6,3,0
28     24,21,18,16,15,11,8,6,3,0
29     27,24,21,18,16,15,11,8,6,3,0
30     27,24,21,18,16,15,11,8,6,3,0
31     27,24,21,18,16,15,11,8,6,3,0
32     30,27,24,21,18,16,15,11,8,6,3,0
33     30,27,24,21,18,16,15,11,8,6,3,0
34     32,30,27,24,21,18,16,15,11,8,6,3,0
35     32,30,27,24,21,18,16,15,11,8,6,3,0
36     35,32,30,27,24,21,18,16,15,11,8,6,3,0
37     35,32,30,27,24,21,18,16,15,11,8,6,3,0
38     35,32,30,27,24,21,18,16,15,11,8,6,3,0
39     35,32,30,27,24,21,18,16,15,11,8,6,3,0
40     38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
41     38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
42     39,35,32,30,27,24,21,18,16,15,11,8,6,3,0
43     40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
44     41,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
45     43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
46     43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
47     43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
48     45,43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0

 n   n^2-1    Trajectories for n^2 - 1, 1 <= n <= 7
--- -------  -----------------------------------------
 1     0      0
 2     3      0
 3     8      6,3,0
 4    15      11,8,6,3,0
 5    24      21,18,16,15,11,8,6,3,0
 6    35      32,30,27,24,21,18,16,15,11,8,6,3,0
 7    48      45,43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0


Ed Jeffery



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