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 A243968 Decimal expansion of 'eta', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1). 2

%I #7 Jan 19 2015 14:48:09

%S 1,4,2,9,8,1,5,4,9,9,9,0,0,9,9,4,5,1,9,7,0,3,9,0,6,4,4,3,7,6,2,7,6,0,

%T 9,3,1,2,6,9,2,3,8,1,5,8,8,4,7,2,5,2,4,2,3,9,5,4,8,2,1,9,4,9,6,9,6,3,

%U 6,2,6,5,4,5,4,3,7,2,8,5,6,8,8,1,1,5,8,3,6,8,9,3,8,4,7,8,1,6,0

%N Decimal expansion of 'eta', 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.

%e 1.42981549990099451970390644376276...

%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); 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

%Y Cf. A006277, A243967 (xi).

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Jun 16 2014

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Last modified August 14 13:35 EDT 2024. Contains 375165 sequences. (Running on oeis4.)