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Search: id:A095708
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| A095708 |
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Tau-functions of the q-discrete Painlev\'{e} 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. 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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4
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| 1, 1, 1, 1, 2, 5, 24, 409, 16648, 2590589, 2837017232, 14797643031281, 589963307907379136, 330879131533072568115765, 1767380481751546168496112185408
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OFFSET
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-2,5
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COMMENT
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Leading order asymptotics of the sequence is log(a(n))~log(2)*n^3/18.
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REFERENCES
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S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics 28 (2002) 119-144.
B. Grammaticos, F. Nijhoff and A. Ramani, Discrete Painlev\'e equations, CRM Series in Mathematical Physics, Ed. R. Conte, Springer-Verlag, New York (1999) 413
A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bulletin of the London Mathematical Society 37 (2005) 161-171.
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LINKS
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A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painleve transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004.
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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.
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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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CROSSREFS
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Cf. A006720.
Sequence in context: A137157 A025134 A076534 this_sequence A120759 A000895 A109306
Adjacent sequences: A095705 A095706 A095707 this_sequence A095709 A095710 A095711
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KEYWORD
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nonn
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AUTHOR
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Andrew Hone (anwh(AT)kent.ac.uk), Jul 07 2004
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