Sequences containing all finite sequences
Franklin T. Adams-Watters
franktaw at netscape.net
Wed Jun 22 07:26:39 CEST 2005
Which sequences contain all finite sequences of non-negative integers as subsequences? (One can also add 1, and look for sequences containing just all finite sequences of positive integers as subsequences. However, the natural examples I found all involved non-negative integers, so that's the way I'm framing the problem.)
Note that this property implies that each sequence occurs infinitely often, since given a sequence A, each of the sequences A,0, A,1, A,2, etc. must occur, and no two can coincide.
I came up with essentially 3 sequences with this property, all tabular sequences. First is A067255, the exponents of the factorizations of n:
1
0,1
2
0,0,1
1,1
etc.
Second is the sequence of with number of zeros following each 1 in the binary representation of n. This is probably the easiest one-to-one correspondence between non-negative integers and finite sequences of non-negative integers. It starts:
0
1
0,0
2
1,0
0,1
0,0,0
3
This sequences was not in the OEIS; I have submitted it.
Third is the digits of n in base factorial:
1
1,0
1,1
2,0
2,1
1,0,0
(see A007623).
This sequence was also not in the encyclopedia, so I submitted it, too.
(One could also reverse each row of any of these, but that seems excessive.)
Are there other sequences with this property in the OEIS - or that should be in the OEIS? Note that all 3 of these are tables, where every finite sequence occurs in a row in an obvious way. It would be nice to find sequences not defined in this way.
--
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
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