The raboter operation transforms a finite word by simultaneously deleting one letter from each maximal run
of identical letters, then concatenating
the remaining runs (Lenormand 2003). If the word
is written as 
,
where adjacent 
differ, 
,
and 
denotes 
copies of the letter 
, then
 |
The empty string is fixed. For a nonempty word, each application reduces its length by the number of its runs, so repeated application reaches the empty string in at most the original length. For example, the word 0011100100 gives 01100, then 10, then the empty string. Leading zeros are retained in this word operation.
For an integer base 
and a nonnegative
integer 
,
let 
be the integer obtained by applying 
to the usual base-
digit string
of 
without leading zeros. The output is read in the same base, with leading zeros discarded
and an empty result assigned the value 0. In particular, 
. In binary, 
gives 
, and 
for 
, 1, 2, ... begins 0, 0, 0, 1, 0, 0, 1, 3, ... (OEIS A318921).
For integers 
and 
, the sum over all 
-digit positive
integers is
 |
The binary case is 
(Zeilberger 2018). Its values
for 
,
2, ... are 1, 4, 14, 46, 146, ... (OEIS A027649).
Biers-Ariel (2019) gives the general-base formula.
To obtain this formula, append a digit to a positive input whose final digit is 
. A different final digit forms a
singleton run and leaves the planed integer
unchanged. Appending another 
instead changes it to 
. Summing over the possible final digits
gives the recurrence equation 
for 
, with 
(Biers-Ariel 2019).
The same run-shortening rule applies to Silverman's sequence. If its terms are denoted by 
, each integer 
occurs 
times in the output. The result begins 2, 3, 4, 4, 5,
5, 6, 6, 6, ... (OEIS A319434).
