Sequence and First
differences use a 3-digit alphabet
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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.
_______________
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...
_______________
Then
Franklin Adams-Watters quickly
answered:
> 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.
Franklin
T. Adams-Watters
_______________
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]