Sequence and First differences use a 3-digit alphabet

[Les suites de cette page n’affichent que des nombres utilisant certains chiffreset pas d’autres.

Toutes les différences entre termes consécutifs desdites suites obéissent à cette même règle]

Hello SeqFans,

here is the idea:

st seq Q = 1 9 18 19  118 119   1118 1119    11118 11119 ...

1st diff =  8 9  1  99   1   999    1    9999     1

We use here only the digits 1, 8 and 9 for Q and Q’s first differences. The pattern is obvious -- and even more obvious in Q’:

st seq Q’= 8 9 18 19  118 119   1118 1119    11118 11119 ...

1st diff =  1 9  1  99   1   999    1    9999     1

Terms of Q and Q’ are positive and monotonically increasing; both seq are infinite. But R isn’t:

st seq R = 3 7  44 47   END

1st diff =  4 37  3   ?

For a 2-digit alphabet, we have:

T1 = 1, 11, 111, 1111, 11111, ... [1st diff. = 10, 100, 1000, 10000, ...]

T2 = 2, 22, 222, 2222, 22222, ... [1st diff. = 20, 200, 2000, 20000, ...]

...

T9 = 9, 99, 999, 9999, 99999, ... [1st diff. = 90, 900, 9000, 90000, ...]

Adding a 3rd term to some T’s seq above brings a bunch of new 3-digit seqs -- for instance:

st seq U = 1 2  12  22   122   222    1222    2222 ...

1st diff =  1 10  10  100   100   1000    1000

st seq U’= 10 11 12  22   122   222 ...

1st diff =   1  1  10  100   100

st seq V = 2 4  24  44   244   444    2444    4444 ...

1st diff =  2 20  20  200   200   2000    2000

st seq V’= 20 22 24  44   244   444 ...

1st diff =   2  2  20  200   200

etc.

Questions:

---------

(a) Are there other such sequences using 3 digits but no zero

(like the seq Q which opens this mail; R was a unfortunate try)?

(b) Could someone compute ALL such 3-digit seq?

(c) What would be the lexicographically first 4-digit seq?

Best,

É.

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A couple of minutes later, I’ve posted this:

> What would be the lexicographically first 4-digit seq?

...with zero, it might be:

st seq Z = 1 2 3  13  23  33   133   233   333 ...

1st diff =  1 1 10  10  10  100   100   100

... but without zero, no idea...

_______________

> What would be the lexicographically first 4-digit seq?

That would be (view with fixed width font):

1,2,3,6,12,13,16,22,23,26,32,33,36,62,63,66,132,133,136,162,163,166,232,233,236,262,263,266,332,333,336,362,363,

1 1 3 6  1  3  6  1  3  6  1  3  26 1  3  66  1   3   26  1   3   66  1   3   26  1   3   66  1   3   26  1   3

366,632,633,636,662,663,666,1332, ...

266  1   3   26  1   3  666

Starting 1,2,3,4 or 1,2,3,5, no next term is possible, so this is minimal. From the above, it should be clear that the sequence can be extended indefinitely.

_______________

I did then post this comment:

Waow, great, Franklin, thanks!

I’ve just found another zeroless 3-digit infinite seq:

1 2 11 12 21 22  121 122  221 222   1221 1222   2221 2222 ...

1 9  1  9  1  99   1   99   1   999    1    999    1

Best,

É.

_______________

Then came this last post, from Douglas McNeil:

There’s also:

[4, 5, 9]

[4, 9,  54, 59,   554, 559,    5554, 5559]

[  5, 45,  5,  495,   5,   4995,    5    ]

Doug

--

Department of Earth Sciences

University of Hong Kong

_______________

Beautiful, Doug!

That’s all for now — many thanks to all who contributed!

Best,

É.

[June 10th, 2010]