[seqfan] Hilbert hotel and first differences -- rooms with label 3k move to 3(k-1)
Eric Angelini
Eric.Angelini at kntv.be
Tue Nov 23 12:30:24 CET 2010
Hello SeqFans,
[if the mail is badly formatted, read its proper disposition here]
[http://www.cetteadressecomportecinquantesignes.com/Hilbert3n.htm]
S = 1,3,2,5,9,4,13,19,6,25,32,7,39,47,8,55,65,10,75,86,11,97,109,12,121,135,14,149,164,15,...
Ask to every term in room 3k to move to room 3(k-1)
The resulting sequence D shows the first differences of S.
(moving rooms have a dot on top and new rooms have a star):
. . . . . . . . . .
S = 1,3,2,5,9,4,13,19,6,25,32,7,39,47, 8,55,65,10,75,86,11,97,109,12,121,135,14,149,164,15,...
D = 2,1,3,4,5,9,6,13,19,7,25,32,8,39,47,10,55,65,11,75,86,12,97,109,14,121,135,15,149,...
* * * * * * * * * *
First differences D of S appear here:
S = 1,3,2,5,9,4,13,19, 6, 25,32, 7, 39,47, 8, 55, 65, 10, 75, 86, 11, 97, 109, 12, 121,135, 14, 149, 164, 15,...
D = 2,1,3,4,5,9, 6, 13,19, 7, 25,32, 8, 39,47, 10, 55, 65, 11, 75, 86, 12, 97, 109, 14, 121, 135, 15, 149,...
-------------------------------------
We wanted S to:
- be a permutation of N
- be the lexicographically first sequence to show those properties
S is a simple concatenation of triplets (a,b,c),(d,e,f),(g,h,i),...
The triplets are linked like this;
-example with (a,b,c) and (d,e,f):
d = b+c
f = the smallest integer not yet present in S
e = d+f
If we start with a=1 we automatically get the whole sequence:
1,3,2,
5,9,4,
13,19,6,
25,32,7,
39,47,8,
55,65,10,
75,86,11,
97,109,12,
121,135,14,
149,164,15,
...
Best,
É.
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