Slowest increasing sequence

[Eric Angelini]

Hello there,

I'm new on the list and would like some help for
something you might have fixed already : how could
one define the constraint : « Slowest increasing
sequence ? »

I've just sent (not yet in the database) this seq :

> 0 1 2 3 4 5 6 7 8 9 21 23 25 27 29 41 43 45 47 49 61 63 65 67 69 81 83 85
87 89 210 301 410 501 610 701 810 901 2101 2103 2105 2107 2109 2121 2123
2125 2127 2129 2141 2143 2145 2147 2149 2161 2163 2165 2167 2169 2181 2183
2185 2187 2189 2301 2303 2305 2307 2309 2321 2323 2325 2327 2329 2341 2343
2345 2347 2349 2361 2363 2365 2367 2369 2381 2383 2385 2387 2389...

With this description :

> Slowest increasing sequence where the digits, taken
> one by one, show the pattern even/odd/even/odd/even...

A friend of mine (Alexandre Wajnberg) wrote me this :

> I have a slowest increasing sequence than yours, Eric :

[YOURS - Eric] :
0 1 2 3 4 5 6 7 8 9  21 23 25 27 29 41 43 45 47 49 61 63 65 67
69 81 83 85 87 89 210 301 410 501 610 701 810 901 2101 2103...

[MINE - Alex] :
0 1 2 3 4 5 6 7 8 10 12 14 16 18 30 32 34 36 38 50 52 54 56 58
70 72 74 76 78 90 92  94  96  98  101 210 301 410 501 610 701
810 901 2101 2103...

> I've just cancelled the lone 9 to get this !

... I was stunned...
... Then had the idea of cancelling something in _his_ sequence
    too ! I came up with this :

> Hello Alex, I'm slower again, thanks to your "cancelling" idea !
I've cancelled your 3-digit numbers and jumped from 98 to 1010 !

  0 1 2 3 4 5 6 7 8 10 12 14 16 18 30 32 34 36 38 50 52 54 56 58
  70 72 74 76 78 90 92  94  96  98
  1010 1012 1014 1016 1018
  1030 1032 1034 1036 1038
  1050 1052 1054 1056 1058
  1070 1072 1074 1076 1078
  1090 1092 1094 1096 1098
  1210 1212 1214 1216 1218
  1230 1232 1234 1236 1238
  1250 1252 1254 1256 1258
  1270 1272 1274 1276 1278
  1290 1292 1294 1296 1298
  1410 1412 1414 1416 1418
  1430 1432 1434 1436 1438
  1450 1452 1454 1456 1458
  1470 1472 1474 1476 1478
  1490 1492 1494 1496 1498
  1610 1612 1614 1616 1618
  1630 1632 1634 1636 1638
  1650 1652 1654 1656 1658
  1670 1672 1674 1676 1678
  1690 1692 1694 1696 1698
  1810 1812 1814 1816 1818
  1830 1832 1834 1836 1838
  1850 1852 1854 1856 1858
  1870 1872 1874 1876 1878
  1890 1892 1894 1896 1898
  3010 3012 3014 3016... etc.

This seq. is slower than Alex' one... How can one define
the "sloweness" of a sequence ? Is there an agreed tool ?

--------

I've encountered the same pb with this sequence of mine :

ID Number: A097912
URL:       http://www.research.att.com/projects/OEIS?Anum=A097912
Sequence:  0,2,3,4,5,6,7,8,9,10,16,27,38,49,50,51,62,73,
           84,90,94,156,278,301,326,478,590,591,623,748,
           750,823,946,1034,1567,2890,2891,3567,4012,4367,
           5890,5891,6234,7012,7345,8690,8723,9456,10345,
           16789,20145,26789,30125...
Name:      Look at the first 10 digits of the sequence :
           they are all different. The same for the next 10.
           And the next 10, etc. This sequence is the slowest
           increasing one with that property.

Ed Murphy (on rec.puzzles) wanted more informations about
my definition of "slowest increasing sequence"...

Regards,
(sorry if this is old hat)
Éric Angelini -- sunny Brussels, Belgium.