# 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.

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