Add and write sequence

[Jacques Tramu]

> From: "Eric Angelini" <Eric.Angelini at kntv.be>

> Start with 1, then a(n) is the smallest integer not used so far
> whose leftmost digits show the sum of a(n-1)'s digits:
> This is the 11-sequence:
>
> 11,2,20,21,3,30,31,4,40,41,5,50,51,6,60,61,7,70,71,8,80,81,9,90,
> 91,10,1,12...
>
> 12 is there because 11 is forbidden; we could thus assign to 11 a
> "pseudo-loop" length -- the number of integers which separate the
> starting "11" from his expected second appearance. Here, this
> pseudo-loop has length 27.
>
> Could someone compute a few pseudo-loop lenghtes (say for n=1 to
> n=200)?
>

Here are the length for n = 1 to 322

{ 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 27, 38, 58, 61, 96, 98, 113, 131, 146, 2, 
27, 38, 49, 157, 169, 197, 406
, 1039, 1468, 2, 27, 38, 42, 49, 63, 161, 192, 199, 223, 2, 27, 38, 52, 55, 
58, 63, 70, 164, 195, 2,27,
 38, 49, 61, 70, 72, 76, 78, 80, 2, 27, 38, 42, 74, 80, 84, 92, 96, 98, 2, 
27, 38, 49, 52, 82, 86, 92, 96, 98, 2, 27, 38, 45, 55, 63, 98, 106, 111, 
113, 2, 27, 32, 49, 58, 61, 111, 117, 129, 131, 28, 49, 63,
70, 74, 115, 125, 140, 143, 144, 38, 49, 52, 70, 74, 80, 125, 129, 143, 182, 
42, 55, 76, 80, 86, 131, 145, 164, 181, 197, 63, 78, 86, 92, 133, 147, 185, 
186, 196, 227, 80, 92, 96, 98, 140, 157, 203, 241, 258,
 259, 98, 106, 111, 157, 161, 208, 262, 269, 290, 291, 111, 117, 163, 188, 
238, 265, 296, 354, 380, 381,
 129, 165, 197, 279, 316, 371, 388, 440, 485, 486, 167, 296, 333, 394, 446, 
489, 510, 562, 607, 608, 299
, 403, 461, 494, 574, 633, 694, 721, 750, 879, 406, 514, 585, 640, 730, 768, 
905, 1002, 1024, 1031, 545,
 647, 750, 774, 1002, 1119, 1133, 1140, 1148, 1168, 672, 774, 1119, 1231, 
1238, 1249, 1260, 1278, 1313,
1335, 777, 1133, 1357, 1373, 1390, 1396, 1414, 1445, 1521, 1528, 1146, 1400, 
1545, 1552, 1573, 1619, 1624, 1647, 1680, 1704, 1454, 1635, 1717, 1747, 
1768, 1839, 1847, 1896, 1984, 2119, 1639, 1749, 2016, 2131,
 2250, 2381, 2434, 2558, 2666, 3158, 1751, 2403, 2579, 2673, 3291, 3817, 
3833, 4200, 4364, 4691, 2716, 4370, 5141, 5699, 5844, 5991, 6376, 6454, 
6938, 6942, 5853, 6474, 7094, 7271, 7350, 7715, 7823, 8266, 829
2, 8438, 223, 255, 312, 514, 545, 1059, 1188, 1194, 1377, 1470, 1635, 1657, 
1770, 1907, 2868, 3169, 3269
, 3319, 3595, 3924, 5150, 5441, 7277}

Starting from n = 323, the lengths are > 10000 .
Regards
JT