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SOMOS MEETS FIBONNACI ROGER C. ALPERIN
 

Summary: SOMOS MEETS FIBONNACI
ROGER C. ALPERIN
Abstract. We describe certain elementary non-linear sequences which
are integer valued and characterize the integral sequences for the special
example xn+1xn-1 = x2
n +1; this is related to the alternate terms of the
Fibonacci sequence.
1. Introduction
We are interested in the sequences generated by the non-linear equation
xn+1xn-1 = x2
n + A
with constant A = 0 with initial values specified for x1, x2. Specifically we
want to know for a given A which integer values of x1 and x2 will give a
sequence consisting only of integers. This is a simplified version of a Somos
sequence; it is known that the sequences generated by this equation generally
have denominators of the form xn-1
1 xn-2
2 [1].
Here are two simple examples: for the A = 1 sequence x1 = 1, x2 = 1, then
the successive terms are 2, 5, 13, . . . , the alternate terms of the Fibonacci

  

Source: Alperin, Roger C. - Department of Mathematics, San Jose State University

 

Collections: Mathematics