Possible to generate sequences given term, length, start and end?

Syntax: \seq[⟨steps before ellipsis⟩,⟨steps after ellipsis⟩]{⟨term label⟩}[⟨step⟩]{⟨first index⟩,...,⟨last index⟩} where every [..] is optional.

\documentclass{scrartcl}

\usepackage{mathtools,amssymb}
\DeclareMathOperator\Skip{Skip}

\usepackage{xparse}

\ExplSyntaxOn

\NewDocumentCommand \seq { o m o m }
 {
  \group_begin:
  \IfValueT {#1}
   {
    \int_set:Nn \l_delrocco_ini_int { \clist_item:nn {#1} {1} }
    \int_set:Nn \l_delrocco_fin_int { \clist_item:nn {#1} {2} }
   }
  \IfValueT {#3}
   {
    \int_set:Nn \l_delrocco_step_int { #3 }
   }
  \delrocco_seq:nn { #2 } { #4 }
  \group_end:
 }

\int_new:N  \l_delrocco_ini_int
\int_set:Nn \l_delrocco_ini_int  { 3 }
\int_new:N  \l_delrocco_fin_int
\int_set:Nn \l_delrocco_fin_int  { 2 }
\int_new:N  \l_delrocco_step_int
\int_set:Nn \l_delrocco_step_int { 1 }


\cs_set_eq:NN \math_sb:n \sb

\cs_new_protected:Npn \delrocco_seq:nn #1 #2
 {
  \int_step_inline:nnnn
   { \clist_item:nn {#2} {1} }
   { \l_delrocco_step_int }
   { \clist_item:nn {#2} {1} + (\l_delrocco_ini_int - 1) * \l_delrocco_step_int }
   { #1 \math_sb:n { ##1 } \, }
  \!\cdots  
  \int_step_inline:nnnn
   { (\l_delrocco_fin_int - 1) * \l_delrocco_step_int }
   { -\l_delrocco_step_int }
   { \l_delrocco_step_int }
   { #1 \math_sb:n { \clist_item:nn {#2} {3} - ##1 } \, }
  #1 \math_sb:n { \clist_item:nn {#2} {3} }
 }
\ExplSyntaxOff

\begin{document}
$\seq[4,4]{a}[2]{5,...,2n}$
\[
  \Skip(L) = \{ \seq[2,1]{a}{1,...,2n-1} \mid \seq{a}{1,...,2n} \in L \}
\]
\end{document}

If you don't want the spaces remove every instance of \, and \!.

With some pain in programming it, the following macros seem to do what you like with a not very complicated syntax.

\seq{<variable>}{<start>..<end>}[<initial>,<final>,<step>]

where <final> and <step> can be omitted and are both set to 1 in this case. If the optional argument is missing, [2,1,1] is implied.

\documentclass{article}
\usepackage{amsmath}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\seq}
 {
  m                               % base letter
  >{\SplitArgument{1}{..}}m       % <start>..<stop>
  >{\SplitArgument{2}{,}}O{2,1,1} % <initial>,<final>,<step>
 }
 {
  \delrocco_seq_print:nnnnnn { #1 } #2 #3
 }

\cs_new_protected:Nn \delrocco_seq_print:nnnnnn
 {
  \tl_if_novalue:nTF { #5 }
   {
    \__delrocco_seq_print:nnnnnn { #1 } { #2 } { #3 } { #4 } { 1 } { 1 }
   }
   {
    \tl_if_novalue:nTF { #6 }
     {
      \__delrocco_seq_print:nnnnnn { #1 } { #2 } { #3 } { #4 } { #5 } { 1 }
     }
     {
      \__delrocco_seq_print:nnnnnn { #1 } { #2 } { #3 } { #4 } { #5 } { #6 }
     }
   }
 }

\cs_new_protected:Nn \__delrocco_seq_print:nnnnnn
 {
  \mathopen{}
  \int_step_inline:nnnn { #2 } { #6 } { #2+(#4-1)*#6 }
   { \mathbin{}\nobreak #1\sb{##1} }
  \mathbin{}\nobreak\dots
  \__delrocco_seq_break:n { #3 }
  \int_step_inline:nnnn { 0-(#5-1)*#6 } { #6 } { 0 }
   {
    \mathbin{}\nobreak
    #1\sb
     {
      \l__delrocco_seq_base_tl
      \int_compare:nTF { \l__delrocco_seq_index_int + (##1) < 0 }
       {
        \int_to_arabic:n { \l__delrocco_seq_index_int + (##1) }
       }
       {
        \int_compare:nF { \l__delrocco_seq_index_int + (##1) = 0 }
         {
          \tl_if_empty:NF \l__delrocco_seq_base_tl { + }
          \int_to_arabic:n { \l__delrocco_seq_index_int + (##1) }
         }
       }
     }
   }
   \mathclose{}
 }

\seq_new:N \l__delrocco_seq_temp_seq
\tl_new:N \l__delrocco_seq_temp_tl
\tl_new:N \l__delrocco_seq_base_tl
\int_new:N \l__delrocco_seq_index_int

\cs_new_protected:Nn \__delrocco_seq_break:n
 {
  \regex_split:nnN { ([\+\-]?[0-9]+\Z) } { #1 } \l__delrocco_seq_temp_seq
  % the last item is empty if there is a base and an index
  \seq_pop_right:NN \l__delrocco_seq_temp_seq \l__delrocco_seq_temp_tl
  \seq_if_empty:NTF \l__delrocco_seq_temp_seq
   {
    \tl_set:Nn \l__delrocco_seq_base_tl {#1}
    \int_set:Nn \l__delrocco_seq_index_int { 0 }
   }
   {
    % the last but one item is the number
    \seq_pop_right:NN \l__delrocco_seq_temp_seq \l__delrocco_seq_temp_tl
    % the base
    \tl_set:Nx \l__delrocco_seq_base_tl { \seq_use:Nn \l__delrocco_seq_temp_seq {} }
    % the index
    \int_set:Nn \l__delrocco_seq_index_int { \l__delrocco_seq_temp_tl }
   }
 }


\ExplSyntaxOff

\begin{document}

$|\seq{a}{1..20}|$

$|\seq{a}{1..2n}[2,2,1]|$

$|\seq{a}{1..2n-1}[3,2,2]|$

$|\seq{a}{1..2n+1}[3,2,2]|$

$|\seq{a}{1..13}[3,2,2]|$

$\operatorname{Skip}(L)=
 \{ \seq{a}{1..2n-1}[2,1,2] \mid \seq{a}{1..2n}[3,2]\text{ is in } L \}
$

$\operatorname{Skip}(L)=
 \{ \seq{a}{1..2n-1}[2,2,2] \mid \seq{a}{1..2n}[3,3]\text{ is in } L \}
$

\end{document}

I didn't allow symbolic calculations for the starting point, but it could be added. No consistency check is done, as it would require quite a lot of algebra.

The space between items is twice \medmuskip, because an empty binary operation is set between any two items. This can be customized.