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A244045 Decimal expansion of the Greenfield-Nussbaum constant, a constant which is the term z(1) in the quadratic recurrence z(0)=1, z(n) = z(n-1)+z(n-2)^2, such that all terms of the bi-infinite sequence z(n) (n = ..., -2, -1, 0, 1, 2, ...) are positive. 0

%I

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

%T 5,8,5,7,0,9,1,1,8,7,0,9,6,8,9,3,0,9,0,0,3,3,8,8,1,1,3,8,7,1,8,2,0,2,

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

%N Decimal expansion of the Greenfield-Nussbaum constant, a constant which is the term z(1) in the quadratic recurrence z(0)=1, z(n) = z(n-1)+z(n-2)^2, such that all terms of the bi-infinite sequence z(n) (n = ..., -2, -1, 0, 1, 2, ...) are positive.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, p. 446.

%H Stephen J. Greenfield and Roger D. Nussbaum, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.3850">Dynamics of a Quadratic Map in Two Complex Variables</a>

%e 1.5078747553927754776624222707709065857...

%t digits = 105; n0 = 10; dn = 10; init[z1_] := (Clear[z]; z[0] = 1; z[1] = z1; z[n_?Positive] := z[n] = z[n-1] + z[n-2]^2; z[n_?Negative] := z[n] = Sqrt[z[n+2] - z[n+1]]); g[z1_?NumericQ, n_] := (init[z1]; Table[z[k], {k, -n, -1}] // Im // Norm); Clear[f]; f[n_] := f[n] = z1 /. FindMinimum[g[z1, n], {z1, 3/2}, WorkingPrecision -> 3*digits][[2]]; f[n0]; f[n = n0 + dn]; While[ RealDigits[f[n], 10, digits] != RealDigits[f[n - dn], 10, digits], Print["n = ", n]; n = n + dn]; RealDigits[f[n], 10, digits] // First

%Y Cf. A000278.

%K nonn,cons

%O 1,2

%A _Jean-Fran├žois Alcover_, Jun 18 2014

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Last modified May 20 23:05 EDT 2022. Contains 353886 sequences. (Running on oeis4.)