[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.
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