Cinquante signes

We look for the digital root of a(n) and concatenate it to a(n). Then iterate. What do successive whole numbers produce? a(1) = 1 , 11, 112, 1124, 11248, 112487, 1124875, 11248751, 112487512, 1124875124, 11248751248, 112487512487, 1124875 124875 , ... The yellow pattern is repeated ad infinitum . a(1) = 2 , 22, 224, 2248, 22487, 224875, 2 248751 , ... The cyan pattern will also be repeated ad infinitum . a(1) = 3 , 33, 336, 3363, 33636, 336363, 3363636, 33636363, 3363636 36 , ... The green pattern will be repeated for ever. a(1) = 4 , 44, 448, 4487, 44875, 448751, 4 487512 , ...  The grey pattern, etc. a(1) = 5 , 55, 551, 5512, 55124, 551248, 5512487, 5 512487 , ...   a(1) = 6 , 66, 663, 6636, 663 63 , ... a(1) = 7 , 77, 775, 7751, 77512, 775124, 7751248, 7 751248 , ... a(1) = 8 , 88, 887, 8875, 88751, 887512, 8 875124 , ... a(1) = 9 , 99, 999, 9999, 9999 9 , ... a(1) = 10 , 101, 1012, 10124, 101248, 1012487, 10 124875 , ... a(1) = 11 , 112, 1124, 11248, 112487, 1 124875 , .....

A histogram is a well-known visual representation of the distribution of quantitative data. What about histonumbers ? We could represent the integers 1234 and 2024 like this:             +---+                    +---+             |   |                    |   |         +---+   +                    +   +         |    |   |                    |   |     +---+   +    +        +---+ ...

The rightmost digit d of a(n) says: "There are exactly  d   even digits immediately after me". This should be the lexicographically earliest sequence  S  of distinct nonnegative numbers with this property. And you know what?  S  was a nightmare to compute by hand! But, as we love Füssli ,  S   was also   a huge pleasure to explore! S   = 0, 1, 21, 23, 2, 20, 3, 200, 5, 4, 22, 40, 7, 6, 24, 42, 60, 9, 8, 26, 44, 62, 80, 10, 11, 25, 202, 201, 27, 204, 2000, 12, 203, 220,  13, 2001, 29, 226, 64, 82, 205, 222, 207, 224, 20001, 41, 43, 240, 14, 20003, 260, 15, 242, 209, 246, 84,  20005 , 262 , 221, 45, 282 , 223, 280, 16, 2002, 225, 402, 227, 244, 20007, 264, 20009, 266, 2022, 229, 286, 2042, 241, 47, 284, 20021, ...  Explanation S   =   0 ,  1 ,  2 1 ,  2 3 ,  2 , 2 0 ,  3 ,  20 0 ,  5 ,  4 , 22, 40 , 7, 6, 24, 42, 60, 9, 8, ... (in blue above, the rightmost digit;...