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arXiv · 1303.5306

How To Generate As Many Somos-Like Miracles as You Wish

Shalosh B. Ekhad·Doron Zeilberger
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Abstract

Jacobi said "man muss immer umkehren". And indeed it takes a genius like Michael Somos to take a specific non-linear recurrence, like a(n)=(a(n-1)a(n-3)+a(n-2)^2)/a(n-4), subject to a(1)=1, a(2)=1, a(3)=1, a(4)=1, and observe that surprise, surprise, they always generate integers. Then it takes other geniuses to actually prove this fact (and the more general so-called Laurent phenomenon). But let's follow Jacobi's advise and go backwards. Rather than try to shoot a target fifty meters away, and most probably miss it, let's shoot first, and then draw the bull'e eye. Then we are guaranteed to be champion target-shooters. So let's take a sequence of integers that manifestly and obviously only consists of integers, and ask our beloved computers to find non-linear recurrences satisfied by the sequence itself, or by well-defined subsequences.

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BibTeXRIS

Shalosh B. Ekhad, Doron Zeilberger. 2013-03-21. How To Generate As Many Somos-Like Miracles as You Wish. https://arxiv.org/abs/1303.5306

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