A101246

[Eric Angelini]

Hello SeqFans,
could someone please compute this -- I've found a cook
by hand in the above seq. and I'm not sure of anything,
now...

A101246 goes like this:

A: 1 3 7 605 607 1446 1538 1608 2545 2618 ...

Write the first differences (D) like that:

A: 1 3 7   605 607   1446  1538  1608   2545  2618 ...
D:  2 4 598   2   839    92    70    937    73

"Push" A into D:

1 2 3 4 7 598 605 2 607 839 1446 92 1538 70 1608 937 2545...

Look now at every successive chunk of 10 digits:

1 2 3 4 7 598 605 2 607 839 1446 92 1538 70 1608 937 2545...
----------------**************------------- ************-

Each underlined chunk shows the 10 digits (0 to 9) only
once.

To build this strictly increasing sequence:
- start with 1
- add the smallest integer which will not contradict the
"10-consecutive-digits" rule immediately.

Example :
start with 1:
- you cannot add 0 (seq. must is strictly increasing)
- you cannot add 1 (as we would have two 1's in the first
  chunk)
- add 2 (smallest available integer)
- We have:
           1 3
            2
- you cannot add now 0, 1, 2 or 3 (same reasons as above)
- add 4 (smallest available integer)
- We have:
           1 3 7
            2 4
- you cannot add now 0, 1, 2, 3, 4, 7. Does 5 fit in?
- Let's try:
            1 3 7 12
             2 4 5
- ouch! We have two 1's and two 2's in the first chunk,
  so 5 is not ok. Let's try 6:
                               1 3 7 13
                                2 4 6
- ouch again! We have two 1's and two 3's in the first
  chunk now... Let's proceed and try 7, 8, 9, 10, 11...
  Because of the 10-digit rule, the first possible can-
  ditate is... 598 !
- So, we now have:
                  1 3 7   605
                   2 4 598
                  ----------*
- Our first chunk is complete. The next one starts with 5.
  We cannot add 0 or 1 but 2 is ok; we get:
                  1 3 7   605 607
                   2 4 598   2
                  ----------*****
- Nothing to add now before 839!
  We have:
                  1 3 7   605 607   1446
                   2 4 598   2   839
                  ----------**********--
- Our second chunk is complete. The third one starts with
  46. Etc.

The cook I've just found this w.-e. appears here; I had
thought the next integer I could add now was 92:
We would have had:
                  1 3 7   605 607   1446  1538
                   2 4 598   2   839    92
                  ----------**********--------
... but what about 83?
Let's try:
                  1 3 7   605 607   1446  1529
                   2 4 598   2   839    83
                  ----------**********--------
This is ok!
So why did I mistake?
Let's go back to the seq. in the OEIS:
                  1 3 7   605 607   1446  1538
                   2 4 598   2   839    92
                  ----------**********--------
The next two available digits to complete the 3rd chunk
are 0 and 7; we cannot add 0, nor 7 alone -- so let's add
70; we get:
                  1 3 7   605 607   1446  1538  1608
                   2 4 598   2   839    92    70
                  ----------**********----------****
... which looks ok.
Let's see again what we would have had adding 83 instead
of 92 previously:
                  1 3 7   605 607   1446  1529
                   2 4 598   2   839    83
                  ----------**********--------
Again, the next two available digits to complete the 3rd
chunk are 0 and 7; we cannot add 0, nor 7 alone -- so let's
add 70; we get:
                  1 3 7   605 607   1446  1529  1599
                   2 4 598   2   839    83    70
                  ----------**********----------****
... ouch! We have two 9's in the 4th chunk! So 70 doesn't
fit (seemingly). And there, I made the mistake to go a step
back and try, instead of 83 --> 84, then 85, 86, ... and
finally 92...

This was wrong, as the rule says: "Add the smallest inte-
ger which will not contradict the "10-consecutive-digits"
rule immediately". The important word is "immediately". In
the example above, 83 should be ok because it doesn't *im-
mediately* contradict the said rule!

What I've understood this w.-e. is that, yes, the last two
digits of the third chunk are 0 and 7, and yes, to add 70
leads to a contradiction, but what about adding 700? Or 701?
Or whatever starting with 70x? This is all what the rule
says! So the seq. in the OEIS must be corrected like this,
from now on:
            1 3 7   605 607   1446  1529    8531
             2 4 598   2   839    83    7002
            ----------**********----------******

Yes, there is nothing less then 7002 to add (I think) accor-
ding to the "10-digits rule"!

As mistakes like that occur very easely by hand, could some-
one compute this, check, and add 20 more terms or so?

My (corrected) seq. would go like this (I think):

1 3 7 605 607 1446 1529 8531 9178 9540 105783 105789 108253
187296 187346 187605 190548 193284 198360 205614 213507 ...

Best,
É.