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In[]:=
EnumerateWolframModelRules[{{1,2}}{{2,2}}]
Out[]=

(For
1
k
n
k
rules, up to a k-term linear recurrence may be involved.)

In[]:=
ParallelMapMonitored[FindLinearRecurrence[(WolframModelTest[#,Automatic]["Sizes"])]&,EnumerateWolframModelRules[{{1,2}}{{2,2}}]]
Out[]=
In[]:=
ParallelMapMonitored[FindLinearRecurrence[(WolframModelTest[#,Automatic]["Sizes"])]&,EnumerateWolframModelRules[{{1,2}}{{3,2}}]]
Out[]=
In[]:=
ParallelMapMonitored[FindLinearRecurrence[(WolframModelTest[#,Automatic]["Sizes"])]&,EnumerateWolframModelRules[{{1,3}}{{2,3}}]]
Out[]=
In[]:=
Length/@%//Counts
Out[]=
18044,21293,336
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