[seqfan] self-describing sequence derived from Markov numbers

wouter meeussen wouter.meeussen at pandora.be
Sat May 14 19:34:50 CEST 2011

```hi sequentialists,

start from the Markov numbers "by generation" :
1 and 2 are bootstrapping it, then comes
{5},
{13, 29},
{34, 194, 433, 169},
{89, 1325, 7561, 2897, 6466, 37666, 14701, 985}
etc.

now, flatten this list, and get the positions of the even integers:
(* they occur in pairs and quadruplets*)
{4, 5, 12, 13, 28, 29, 32, 33, 38, 39, 40, 41, 46, 47, ...
then get the lengths of the runs and decrease by 1:

{1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3,
1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1,
3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3,
1,
1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 1, 1,
1,
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3,
1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1,
1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1,
1,
1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1,
3,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1,
1,
3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1,
1,
1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1,
1,
1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1,
1,
1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
3,
1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1,
1,
1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1,
1,
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3,
3,
3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, ...

get the positions of the 3's :
{5, 13, 25, 29, 34, 35, 36, 41, 52, 60, 65, 76, 81, 88, 93, 94, 95, 100,
111,
123, 128, 147, 152, 167, 172, 183, 188, 195, 200, 201, 202, 207, 218, 222,
227, 228, 229, 234, 245, 250, 261, 266, 267, 268, 273, 274, 275, 280, 281,
282, 287, 298, 303, 314, 319, 320, 321, 326, 337, 342, 361, 366, 381, 386,
397, 402, 409, 414, 415, 416, 421, 432, ...}

and this, surprisingly, again has runs of lengths:
{1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3,
1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1,
3, 1, 1, ...}
get the positions of the 3's :
{5, 13, 25, 29, 34, 35, 36, 41, 52, ...}