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Decimal expansion of (1 + sqrt(3))/2, unique positive root of x^2 - x - 1/2.
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%I #39 Nov 12 2024 08:16:56

%S 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,

%T 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,

%U 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

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

%C 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}.

%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

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>

%F Equals 1/2 + Sum_{n>=0} ((-1)^(n + 1)*binomial(2*n, n))/(2^(3*n + 1/2)*(2*n - 1)). - _Antonio GraciĆ” Llorente_, Nov 11 2024

%e 1.3660254037844386467637231707529361834714026269051903140279...

%t RealDigits[(1 + Sqrt[3])/2, 10, 120][[1]] (* _Amiram Eldar_, Jun 21 2023 *)

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

%o (PARI) polrootsreal(2*x^2-2*x-1)[2] \\ _Charles R Greathouse IV_, Jan 26 2023

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

%Y Cf. A001622, A174968.

%K nonn,cons

%O 1,2

%A _M. F. Hasler_, Oct 29 2020