Making calculations using counter

A combination of forloop and calc:

\documentclass[10pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{calc}
\usepackage{forloop}
\begin{document}

\hsize=8cm
\newcounter{k}\newcounter{j}%
\begin{align}
%forloop[ step ]{ counter }{ initial value }{ condition }{ code }
\nonumber
\forloop{k}{1}{\value{k} < 11}{
\setcounter{j}{2*\value{k}-1}
\\ y_{\arabic{k}}&=A_{\arabic{k},\arabic{j}} x_{\arabic{j}}
\addtocounter{j}{1}
+A_{\arabic{k},\arabic{j}} x_{\arabic{j}}
}
\end{align}

\end{document}

Here another approach using l3int:

\documentclass{article}
\usepackage{amsmath}
\usepackage{xparse} %dt. Silbentrennung
\ExplSyntaxOn
\cs_generate_variant:Nn \int_set:Nn { Nx }
\NewDocumentCommand \Formula { O{1} O{1} m }
 {
  \int_step_inline:nnnn
     { #1 }%initial value
     { #2 }%step
     { #3 - #2 }%final value
     { \formula_func_aux:n { ##1 } \\ }%code
  \formula_func_aux:n { #3 } %last line without \\
 }

\cs_new:Npn  \formula_func_aux:n #1
 {
  \int_gset:Nn \g_tmpa_int { #1 }
      y \sb{ \int_use:N \g_tmpa_int } &=
      \int_gset:Nn \g_tmpb_int { 2 * \g_tmpa_int -1  }
      A \sb{ \int_use:N \g_tmpa_int , \int_use:N \g_tmpb_int }
      x \sb{ \int_use:N \g_tmpb_int }
     +
      \int_gset:Nn \g_tmpb_int { 2 * \g_tmpa_int }
      A \sb{ \int_use:N \g_tmpa_int , \int_use:N \g_tmpb_int }
      x \sb{ \int_use:N \g_tmpb_int } 
 }

\ExplSyntaxOff
\begin{document}
\begin{align}
\Formula{2}
\end{align}

\begin{align}
\Formula[1][2]{6}
\end{align}
\end{document}

This also uses no counters but is perhaps a bit simpler than jfbu's but is the same idea.

\documentclass{article}

\usepackage{amsmath}


\newcommand\y[1]{%
y_{\the#1}&=A_{\the#1\,\the\numexpr2*#1-1\relax}
      +A_{\the#1\,\the\numexpr2*#1\relax}
        x_{\the\numexpr2*#1\relax}}

\def\lp#1#2#3{\ifnum#1=#3\relax\stoplp\fi#2{\numexpr#1\relax}\\\lp{\numexpr#1+1\relax}#2{#3}}

\def\stoplp#1\\#2#3#4#5{#1}

\begin{document}

\begin{align}
\lp1\y{10}
\end{align}

\end{document}