A389134 Decimal expansion of the icosahedral root of 2.

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

Offset

1,2

Comments

The real solution to x*(5*x^2 - 5*x + 2)/2 = 2. Minimum polynomial 5*x^3-5*x^2+2*x-4.

Links

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Index entries for algebraic numbers, degree 3.

Formula

Equals ((3 * sqrt(6945) + 250)^(1 / 3) / 5^(2 / 3) - 1 / (5^(1 / 3) * (3 * sqrt(6945) + 250)^(1 / 3)) + 1) / 3.

Example

1.213585191148275805896983813...

Mathematica

N[((3 Sqrt[6945] + 250)^(1/3)/5^(2/3) - 1/(5^(1/3) (3 Sqrt[6945] + 250)^(1/3)) + 1)/3, 105]

Prog

(PARI) ((3*sqrt(6945)+250)^(1/3)/5^(2/3)-1/(5^(1/3)*(3*sqrt(6945)+250)^(1/3))+1)/3

Crossrefs

Cf. A006564, A002193 (square root of 2), A222133 (triangular root of 2), A188934 (hexagonal root of 2).

Platonic figurate roots of 2: A002580 (cube root), A388148 (tetrahedral root), A389133 (octahedral root), this sequence (icosahedral root), A389135 (dodecahedral root).

Sequence in context: A135017 A070168 A246646 * A198094 A372660 A263047

Adjacent sequences: A389131 A389132 A389133 * A389135 A389136 A389137

Keyword

nonn,cons

Author

Kelvin Voskuijl, Sep 24 2025

Status

approved