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%I #10 Jan 19 2015 14:48:01
%S 1,3,5,0,5,0,6,1,2,5,1,3,1,1,7,1,5,3,0,3,3,1,8,3,7,6,7,7,2,2,6,2,4,1,
%T 5,9,7,2,5,2,3,0,6,9,8,0,3,1,3,0,1,9,2,5,5,8,6,0,9,7,8,4,0,6,1,6,4,5,
%U 0,7,4,0,0,8,8,8,8,1,5,1,3,5,8,8,9,8,3,4,8,3,5,5,6,8,5,1,5,1,1
%N Decimal expansion of 'xi', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, pp. 445-446.
%F q(n) = floor(xi^(phi^n)*eta^((1-phi)^n)) where phi is the golden ratio (1+sqrt(5))/2 and eta is A243968.
%e 1.3505061251311715303318376772262415972523...
%t digits = 99; n0 = 5; dn = 5; Clear[q]; q[0] = q[1] = 1; q[n_] := q[n] = q[n - 2] (q[n - 1] + 1); xi[n_] := xi[n] = ((q[n] - 1)^(1/2*( Sqrt[5] - 1))*(q[n + 1] - 1))^((1/2*( Sqrt[5] - 1))^n/Sqrt[5]); xi[n0]; xi[n = n0 + dn]; While[RealDigits[xi[n], 10, digits + 10] != RealDigits[xi[n - 5], 10, digits + 10], Print["n = ", n]; n = n + dn]; RealDigits[xi[n], 10, digits] // First
%Y Cf. A006277, A243968 (eta).
%K nonn,cons
%O 1,2
%A _Jean-François Alcover_, Jun 16 2014
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