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A244045
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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.
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0
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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, 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, 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
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OFFSET
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1,2
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, p. 446.
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LINKS
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EXAMPLE
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1.5078747553927754776624222707709065857...
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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