# [seqfan] Extended autobiographical numbers

Éric Angelini bk263401 at skynet.be
Wed Aug 14 16:50:32 CEST 2019

```Hello SeqFans,
We all know the autobiographical numbers 1210, 2020, 21200,
3211000, 42101000, 521001000, 6210001000 presented here:
https://oeis.org/A046043.

What if, instead of the digits (0 to 9), we wanted to (self)
document the number of <Digit-Substrings> "0" to "n" in a(n)?

[I am almost sure this is old hat, forgive me]

DS = 0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17...
a(1) 1  2  1  0
a(2) 2  0  2  0
a(3) 2  1  2  0  0
a(4) 3  2  1  1  0  0  0
a(5) 4  2  1  0  1  0  0  0
a(6) 5  2  1  0  0  1  0  0  0
a(7) 6  2  1  0  0  0  1  0  0  0
----------------------------
a(8) 5  3  1  1  0  1  0  0  0  0  2             (2 x "10" in a(8)
a(9) 6  2  2  0  0  0  1  0  0  0  1             (1 x "10" in a(9)
a(.) 5  4  1  0  1  1  0  0  0  0  2  1          (2 x "10", 1 x "11")
a(.) 6  4  0  1  1  0  1  0  0  0  3  1  0       (3x10, 1x11, 0x12)
a(.) 7  4  0  1  1  0  0  1  0  0  3  1  0  0           (same + 0x13)
a(.) 8  4  0  1  1  0  0  0  1  0  3  1  0  0  0        (same + 0x14)
a(.) 9  4  2  0  1  0  0  0  0  1  2  1  1  0  0  0  0   (2x10, etc.)
-------------------------------------------------
I think we can go (infinitely?) further:

a(.)10  4  0  1  1  0  0  0  0  0  3  1  0  0  0
a(z)11  6  0  0  1  0  1  0  0  0  4  1  0  0  0  0  1  0

This last "proposition" must be read (as you all understood):
« There are
11 strings "0" in a(z),
6 strings "1"
0 strings "2"
0 string  "3"
1 string  "4"
0 string  "5"
1 string  "6"
0 string  "7", "8" and "9"
4 strings "10"
1 string  "11"
0 string  "12", "13", "14" and "15"
1 string  "16" (visible in "11 6")
and 0 string  "17" ».

Best,
É.

```