Somos 7 Sequence

Define a sequence with:

a(0)=a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=1 , and a(n)=(a(n-1)*a(n-6)+a(n-2)*a(n-5)+a(n-3)*a(n-4))/a(n-7) if n > 6 .

The first few terms are: 1,1,1,1,1,1,1,3,5,9,17,41,137,769,1925,7203,34081 and is listed as A006723 in Sloane's Online Encyclopedia of Integer Sequences (OEIS). For more information about this and related sequences, see article "SomosSequence" in Eric Weisstein's MathWorld.

The sequence is one of a large class of sequences of numbers that satisfy a non-linear recurrence relation depending on previous terms. It is also one of the class of sequences which can be computed from a theta series, hence I call them theta sequences. Here is the details:

Fix the following seven constants:

c1 = 0.920636759658310327248345..., c2 = 1.065943011293868830521869...,
c3 = 0.124419491689603332761100..., c4 = 0.064987692264348244718361...,
c5 = 1.243718669048843700581632..., c6 = 1.237840563909471899772811..., and
c7 = 0.945850084294314824517469... . Consider the doubly indexed series:

f(x,y,s) = c1*c2^(x*x)*sum(k2, s^k2*sum(k1, g(k1,k2,x,y))) , where
g(k1,k2,x,y) = c3^(k1*k1)*c4^(k2*k2)*c5^(k1*k2)*cos(c6*k1*x+c7*k2*y) .
Here both sums range over all integers. Then the sequence defined by

a(n) = k*f(n-3,n-3,(-1)^n), where k = .6^.25 if n even otherwise 1,

is the Somos 7 sequence.  I announced this on August 26, 2011.

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Last Updated Sep 07 2022
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