I don't know if this is the whole story, or page 1 of a longer paper. --Rich ----- Forwarded message from jamespropp@GMAIL.COM ----- Date: Thu, 2 Feb 2012 00:36:08 -0800 From: James Propp <jamespropp@GMAIL.COM> Reply-To: The Robbins Forum <ROBBINS@listserv.uml.edu> Subject: [ROBBINS] posted on behalf of Michael Somos To: ROBBINS@listserv.uml.edu ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Trilinear Recursion Michael Somos somos@cis.csuohio.edu It has been known for some time that the recursion A(n) = (A(n-1) + C1) / A(n-2) (of which C1=1 is the Lyness 5-cycle) has a solution in terms of a general Somos-5 sequence a(n) as follows A(n) = (a(n+2) * a(n-3)) / (a(n) * a(n-1) * C2) 1 + A(n) = (a(n+1) * a(n-2) * C3) / (a(n) * a(n-1) * C2) for some constansts C2, C3. There are similar kinds of examples. One of the simplest is the following. If A(n) = (A(n-1) + A(n-2)) / A(n-3), A(1) = A(2) = A(3) = 1, then A(n) = (a(n) * a(n+7)) / (a(n+3) * a(n+4)) where a(n)=(a(n-1)*a(n-6)*a(n-8) + a(n-2)*a(n-4)*a(n-9)) / (a(n=5)*a(n-10)), a(1) = a(2) = ... = a(10) = 1. which is now sequence A205303 in the OEIS at oeis.org. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ----- End forwarded message ----- ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Trilinear Recursion Michael Somos somos@cis.csuohio.edu It has been known for some time that the recursion A(n) = (A(n-1) + C1) / A(n-2) (of which C1=1 is the Lyness 5-cycle) has a solution in terms of a general Somos-5 sequence a(n) as follows A(n) = (a(n+2) * a(n-3)) / (a(n) * a(n-1) * C2) 1 + A(n) = (a(n+1) * a(n-2) * C3) / (a(n) * a(n-1) * C2) for some constansts C2, C3. There are similar kinds of examples. One of the simplest is the following. If A(n) = (A(n-1) + A(n-2)) / A(n-3), A(1) = A(2) = A(3) = 1, then A(n) = (a(n) * a(n+7)) / (a(n+3) * a(n+4)) where a(n)=(a(n-1)*a(n-6)*a(n-8) + a(n-2)*a(n-4)*a(n-9)) / (a(n=5)*a(n-10)), a(1) = a(2) = ... = a(10) = 1. which is now sequence A205303 in the OEIS at oeis.org. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++