A095708 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.

1, 1, 1, 1, 2, 5, 24, 409, 16648, 2590589, 2837017232, 14797643031281, 589963307907379136, 330879131533072568115765, 1767380481751546168496112185408, 144316081261864174415255171573551803529, 331532406835855871870984528299440847142815328384

Offset

-2,5

Comments

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.

References

B. Grammaticos, F. Nijhoff and A. Ramani, Discrete Painlevé equations, CRM Series in Mathematical Physics, Ed. R. Conte, Springer-Verlag, New York (1999) 413.

Links

Robin Visser, Table of n, a(n) for n = -2..39

S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics 28 (2002) 119-144.

A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004.

Formula

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

Maple

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;

Mathematica

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}];
Table[t[n], {n, -2, nmax}] (* Jean-François Alcover, Aug 16 2018, from Maple *)

Crossrefs

Cf. A006720, A014125.

Sequence in context: A340772 A370066 A076534 * A120759 A326971 A000895

Adjacent sequences: A095705 A095706 A095707 * A095709 A095710 A095711

Keyword

nonn

Author

Andrew Hone, Jul 07 2004

Extensions

More terms from Robin Visser, Aug 20 2023

Status

approved