A396258 - OEIS

A396258

Decimal expansion of the largest root to 64*x^6 - 672*x^4 + 1764*x^2 - 189 = 0.

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Negated real part of the Gauss sum tau(chi) = Sum_{a=0..6} chi(a)*exp(2*Pi*i/7), where chi is the Dirichlet character modulo 7 such that chi(3) = exp(2*Pi*i/6). Note that tau(chi) is a root to x^12 + 497*x^6 + 117649 = 0.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for algebraic numbers, degree 6.

EXAMPLE

2.44013335834553767898...

MATHEMATICA

First[RealDigits[Root[64*#^6 - 672*#^4 + 1764*#^2 - 189 &, 6], 10, 100]] (* Paolo Xausa, May 21 2026 *)

PROG

(PARI) solve(x=2.4, 2.5, 64*x^6 - 672*x^4 + 1764*x^2 - 189)

CROSSREFS

Gauss sums of nontrivial Dirichlet characters modulo 7:

Characters | Gauss sum