Ternary analogue of A094913?

[Giovanni Resta]

Jonathan Post wrote:
> What is the ternary analogue of  A094913 Maximum number of distinct
> substrings of any binary string of length n?
>
> a(n) = Maximum number of distinct substrings of any ternary string of length n.
>
> Cf. A007089  Numbers in base 3.
>
> By hand, it seems to begin:
>
> n   a(n)  example
> 1   1       either 1 or 2 has only itself as substring
> 2   3       12 contains 1, 2, 12
> 3   6       120 contains 1, 2, 0, 12, 20, 120
> 4   9       1201 contains 1,2,0,12,20,01,120,201,1201
> 5   13     12010 has 1,2,0,12,20,01,10,120,201,010,1201,2010,12010
> 6   18     120102 has 1,2,0,12,20,01,10,02,120,201,010,102,1201,2010,0102,12010,
> 20102,120102
> 7  24?
I have not the the time to study the subject, but I offer some values:
(use a fixed width font to visualize correctly the table)
---------------------------------------
n  a(n)  lexicograf. first such string
---------------------------------------
1     1   0
2     3   01
3     6   012
4     9   0012
5    13   00102
6    18   001020
7    24   0010211
8    31   00102112
9    39   001021120
10   48   0010211220
11   57   00010211220
12   67   000100211220
13   78   0001002011221
---------------------------------------
I hope this helps,
giovanni.