0,3

A second order recurrence with the Laurent property. This property is satisfied by any second order recurrence of the form a(n+2)=f(a(n+1))/a(n) where f is a polynomial of the form f(x)=x^m*p(x) with m a positive integer and p arbitrary. So if p has integer coefficients and a(0)=a(1)=1 then a(n) is an integer for all n.

As n tends to infinity, log(log(a(n)))/n tends to log((3+sqrt(5))/2) or about 0.962.

The Laurent property is satisfied by any second order recurrence of the form a(n+2)=f(a(n+1))/a(n) where f is a polynomial of the form f(x)=x^m*p(x) with m a positive integer greater than or equal to 2 and p arbitrary. In that case a(0)=a(1)=1 generates a sequence of integers, and the ratios a(n+1)/a(n) and a(n+1)*a(n-1)/a(n)^2 are integers for all n. - Andrew Hone (anwh(AT)kent.ac.uk), Dec 12 2005

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

a[0]:=1; a[1]:=1; f(x):=x^3+x^2; for n from 0 to 8 do a[n+2]:=simplify(subs(x=a[n+1], f(x))/a[n]) od; s[3]:=ln(2^2*3); s[4]:=ln(2^3*3^2*13); for n from 3 to 10000 do s[n+2]:=evalf(3*s[n+1]+ln(1+exp(-s[n+1]))-s[n]): od: print(evalf(ln(s[10002])/(10002))): evalf(ln((3+sqrt(5))/2)); # s[n]=ln(a[n]); ln(s[n])/n converges slowly to 0.962...

Sequence in context: A061149 A129933 A064320 this_sequence A058975 A057120 A112512

Adjacent sequences: A112370 A112371 A112372 this_sequence A112374 A112375 A112376

nonn

Andrew Hone (anwh(AT)kent.ac.uk), Dec 02 2005