OFFSET
1,2
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, pp. 445-446.
FORMULA
q(n) = floor(xi^(phi^n)*eta^((1-phi)^n)) where phi is the golden ratio (1+sqrt(5))/2.
EXAMPLE
1.42981549990099451970390644376276...
MATHEMATICA
digits = 99; n0 = 5; dn = 5; Clear[q]; q[0] = q[1] = 1; q[n_] := q[n] = q[n - 2] (q[n - 1] + 1); eta[n_] := eta[n] = ((q[n] - 1)^(-1/2 - Sqrt[5]/2)*(q[n + 1] - 1))^(-(1/((1/2*(1 - Sqrt[5]))^n*Sqrt[5]))); eta[n0]; eta[n = n0 + dn]; While[RealDigits[eta[n], 10, digits + 10] != RealDigits[eta[n - 5], 10, digits + 10], Print["n = ", n]; n = n + dn]; RealDigits[eta[n], 10, digits] // First
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 16 2014
STATUS
approved