A113786 Decimal expansion of the solution to the equation sqrt(x) + sqrt(x+1) + sqrt(x+2) = sqrt(9*x+8).

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

Offset

1,1

Links

Table of n, a(n) for n=1..100.

Index entries for algebraic numbers, degree 3.

Formula

Equals (1/24)*(2*(3*(90 + sqrt(1509)))^(1/3)+ 2*(3*(90 - sqrt(1509)))^(1/3)-9).

Minimal polynomial: 512*x^3 + 576*x^2 - 200*x - 289. - Amiram Eldar, Jun 08 2026

Equals -3/8 + sqrt(13/3) * cosh(arccosh(30*sqrt(3)/13^(3/2))/3)/2. - Vaclav Kotesovec, Jun 08 2026

Example

0.67819352379110344286342528025892014001502874941365....

Mathematica

RealDigits[n/.FindRoot[Sqrt[n ]+Sqrt[n+1]+Sqrt[n+2]==Sqrt[9n+8], {n, 1}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Apr 25 2018 *)

Prog

(PARI) polrootsreal(512*x^3 + 576*x^2 - 200*x - 289)[1] \\ Charles R Greathouse IV, May 19 2026

Crossrefs

Cf. A114458 (floor(sqrt(n)+sqrt(n+1)+sqrt(n+2))).

Sequence in context: A284362 A394809 A229177 * A004487 A274137 A010500

Adjacent sequences: A113783 A113784 A113785 * A113787 A113788 A113789

Keyword

cons,nonn,changed

Author

Zak Seidov, Jan 20 2006

Status

approved