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A095708
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Tau-functions of the q-discrete Painlevé I equation, f(n+1) = (A*q^n*f(n) + B)/(f(n)^2*f(n-1)), for q=2 and A=B=1, with f(n) = a(n+1)*a(n-1)/a(n)^2.
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4
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1, 1, 1, 1, 2, 5, 24, 409, 16648, 2590589, 2837017232, 14797643031281, 589963307907379136, 330879131533072568115765, 1767380481751546168496112185408, 144316081261864174415255171573551803529, 331532406835855871870984528299440847142815328384
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OFFSET
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-2,5
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COMMENTS
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Leading order asymptotics of the sequence is log(a(n))~log(2)*n^3/18.
In general a(n) is a polynomial in q; here evaluated at the value q=2. For q=1 it is the Somos-4 sequence.
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REFERENCES
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B. Grammaticos, F. Nijhoff and A. Ramani, Discrete Painlevé equations, CRM Series in Mathematical Physics, Ed. R. Conte, Springer-Verlag, New York (1999) 413.
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LINKS
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FORMULA
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a(n) = (2^(n-2)*a(n-1)*a(n-3) + a(n-2)^2)/a(n-4); a(-2)=a(-1)=a(0)=a(1)=1.
0 = a(n+6)*a(n+2)*a(n+1) - 4*a(n+5)*a(n+4)*a(n) + 4*a(n+5)*a(n+2)*a(n+2) - a(n+4)*a(n+4)*a(n+1) for all n in Z. - Michael Somos, Jan 21 2014
0 = a(n+5)*a(n+3)*a(n+1)*a(n+1) - 2*a(n+4)*a(n+4)*a(n+2)*a(n) + 2*a(n+4)*a(n+2)^3 + a(n+3)^3*a(n+1) for all n in Z. - Michael Somos, Jan 21 2014
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MAPLE
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t[0]:=1; t[1]:=1; t[ -2]:=1; t[ -1]:=1; alpha:=1; beta:=1; for n from 0 to 12 do t[n+2]:=simplify((alpha*2^n*t[n+1]*t[n-1]+beta*t[n]^2)/t[n-2]): od;
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MATHEMATICA
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nmax = 12; t[-2] = t[-1] = t[0] = t[1] = 1;
Do[t[n+2] = (2^n*t[n+1]*t[n-1] + t[n]^2)/t[n-2], {n, 0, nmax}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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