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A358187
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Decimal expansion of the positive real root r of x^4 + 2*x^3 - 1.
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1
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7, 1, 6, 6, 7, 2, 7, 4, 9, 2, 8, 2, 2, 8, 6, 6, 3, 8, 4, 2, 4, 7, 3, 9, 3, 0, 1, 4, 3, 2, 5, 5, 6, 1, 8, 3, 9, 2, 1, 5, 5, 1, 3, 7, 6, 0, 2, 9, 8, 6, 1, 6, 4, 6, 6, 7, 8, 9, 4, 5, 6, 8, 0, 2, 4, 2, 1, 4, 7, 4, 9, 0, 0, 7, 3, 3, 8, 7
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OFFSET
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0,1
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COMMENTS
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The roots of x^4 + 2*x^3 - 1 are X = Y - 1/2, where Y are the roots of y^4 - (3/2)*y^2 + y - 19/16.
The other real root is -A358188, and the complex roots are -0.3048767044... + 0.7545291731...*i and its complex conjugate. The formula for the first complex root is obtained by replacing 3 and the two 4s with their negative values in the formula for r given below, while using the unchanged u.
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LINKS
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FORMULA
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r = (1/2)*((-3 - 4*u + sqrt(9 - 16*u^2 + 4*sqrt(8*u + 6)))/sqrt(8*u + 6) - 1), with u = (((3/4)*(-9 + sqrt(129)))^(1/3) + w1*((3/4)*(-9 - sqrt(129)))^(1/3) - 3/4)/3 = -0.06738537990..., where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1. Or with hyperbolic functions u = -(2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))) - 1/4.
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EXAMPLE
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0.716672749282286638424739301432556183921551376029861646678945680242147...
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MATHEMATICA
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RealDigits[x /. FindRoot[x^4 + 2*x^3 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)
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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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