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A358189
Decimal expansion of the positive real root r of x^4 + 2*x - 1.
1
4, 7, 4, 6, 2, 6, 6, 1, 7, 5, 6, 2, 6, 0, 5, 5, 5, 0, 3, 2, 9, 4, 1, 3, 2, 0, 9, 8, 9, 4, 9, 3, 1, 4, 1, 2, 6, 6, 7, 3, 6, 1, 3, 6, 5, 9, 1, 9, 4, 7, 8, 5, 2, 2, 3, 4, 9, 5, 6, 5, 6, 3, 2, 6, 1, 1, 4, 3, 1, 1, 1, 3, 0, 2, 5, 7, 8, 6
OFFSET
0,1
COMMENTS
The other real root is -A358190. One of the complex roots is 0.4603551884... + 1.1393176803...*i, given by (sqrt(2)*u + sqrt(-2*u^2 - 2*sqrt(2*u)))/(2*sqrt(u)), where u is given in the formula below. The other complex root is its conjugate (with a minus sign for the second sqrt).
FORMULA
r = (-sqrt(2)*u + sqrt(-2*u^2 + 2*sqrt(2*u)))/(2*sqrt(u1)), where u = (((3/4)*(9 + sqrt(129)))^(1/3) + w1*((3/4)*(9 - sqrt(129)))^(1/3))/3 = 0.4238537990..., and w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3), is one of the complex roots of x^3 - 1. Alternatively u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))).
EXAMPLE
0.47462661756260555032941320989493141266736136591947852234956563261143...
MATHEMATICA
RealDigits[x /. FindRoot[x^4 + 2*x - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)
CROSSREFS
Cf. A358190.
Sequence in context: A376549 A019735 A202501 * A195362 A106739 A112677
KEYWORD
nonn,cons,easy
AUTHOR
Wolfdieter Lang, Dec 07 2022
STATUS
approved