A332133 Decimal expansion of (1 + sqrt(3))/2, unique positive root of x^2 - x - 1/2.

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

Offset: 1

Author

M. F. Hasler, Oct 29 2020

Keywords

• nonn
• cons

Comments

Also, max {a, b} where {a,b} is the unique solution of a + b = 1 and a^2 + b^2 = 2 (implying also ab = -1/2 and a^3 + b^3 = 5/2 without solving for a, b). See A332122 for a generalization to 3 variables {a, b, c}.

This is a non-integer element of the quadratic number field Q(sqrt(3)) with the given monic minimal polynomial. The other negative root is -(-1 + sqrt(3))/2 = - A152422. - Wolfdieter Lang, Aug 30 2022

Examples

			1.3660254037844386467637231707529361834714026269051903140279...
		

Links

• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A152422 (this - 1 = (sqrt(3)-1)/2), A010527, A332122 (analog for 3rd degree).

Cf. A001622, A174968.

Programs

Mathematica

RealDigits[(1 + Sqrt[3])/2, 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

PARI

localprec(111); digits(solve(a=0,2,a^2-a-1/2)\.1^99)

PARI

polrootsreal(2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Jan 26 2023

Formula

Equals 1/2 + Sum_{n>=0} ((-1)^(n + 1)*binomial(2*n, n))/(2^(3*n + 1/2)*(2*n - 1)). - Antonio Graciá Llorente, Nov 11 2024