[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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