​
EnumerateWolframModelRules[{{1,2}}{{2,2}}]
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(For
1
k

n
k
rules, up to a k-term linear recurrence may be involved.)
​

ParallelMapMonitored[FindLinearRecurrence[(WolframModelTest[#,Automatic]["Sizes"])]&,EnumerateWolframModelRules[{{1,2}}{{2,2}}]]
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ParallelMapMonitored[FindLinearRecurrence[(WolframModelTest[#,Automatic]["Sizes"])]&,EnumerateWolframModelRules[{{1,2}}{{3,2}}]]
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ParallelMapMonitored[FindLinearRecurrence[(WolframModelTest[#,Automatic]["Sizes"])]&,EnumerateWolframModelRules[{{1,3}}{{2,3}}]]
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Length/@%//Counts
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18044,21293,336
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