Look-and-say sequence: roman numerals edition
Perl, 49 bytes

Includes +1 for -p

Run with the 0-based index on STDIN, e.g.

ecce.pl &lt;&lt;&lt; 14


ecce.pl:

#!/usr/bin/perl -p
s,(.)\1*,$&amp;/$1%182 .$1,eg for($_=/$/)x$`;y;19;IV


Magic formulas are so magic.

Normally I would use ($_=//)x$' to make the loop control one byte shorter, but scoring on this site gives that a handicap of 2 so it ends up 1 byte longer. On older perls you can drop the space before for. Some versions of perl force you to add a final ; to close the transliteration. But what is given above is the code that works on my system.

Explanation

Working backwards from solution to code:

The string transformations we need:

I     -&gt; II
II    -&gt; III
III   -&gt; IIII
IIII  -&gt; IVI
IIIII -&gt; VI

V     -&gt; IV
VV    -&gt; IIV


Each replacement ends with the repeated character. I will get a sequence of the same characters using regex /(.)\1*/, so this can be done by appending $1. The part before the -&gt; is in $&amp;. With that I still need:

I     -&gt; I
II    -&gt; II
III   -&gt; III
IIII  -&gt; IV
IIIII -&gt; V

V     -&gt; I
VV    -&gt; II


Write I as 1 and V as 9:

1     -&gt; 1
11    -&gt; 11
111   -&gt; 111
1111  -&gt; 19
11111 -&gt; 9

9     -&gt; 1
99    -&gt; 11


By dividing the part before -&gt; by the repeated digit this becomes:

1     -&gt; 1
11    -&gt; 11
111   -&gt; 111
1111  -&gt; 19
11111 -&gt; 9

1     -&gt; 1
11    -&gt; 11


So now the original repeated V is not an exception anymore. So I want an expression that makes this happen:

1     -&gt; 1
11    -&gt; 11
111   -&gt; 111
1111  -&gt; 19
11111 -&gt; 9


And this can be done by a simple modulo 182:

1     % 182 = 1
11    % 182 = 11
111   % 182 = 111
1111  % 182 = 19
11111 % 182 = 9


(this even gets IIIIII to VI right though it isn't needed here)

All that is left is initializing the working variable to 1 for index 0, repeat this transformation in a loop and at the end replace 1 by I and 9 by V

1, 9 and 182 is the only parameter combination for which this simple formula works.