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Decimal expansion of (-1 + sqrt(13))/6 = A223139/3.
4

%I #15 Dec 31 2023 00:28:09

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

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

%U 4,4,9,1,3,8,2,1,6,8

%N Decimal expansion of (-1 + sqrt(13))/6 = A223139/3.

%C This constant r, an algebraic integer of the quadratic number field Q(13), is the positive root of its monic minimal polynomial x^2 + x/3 - 1/3. The negative root is -(1 + sqrt(13))/6 = -A209927/3 = -(A188943 - 1).

%C r^n = A052533(-n) + A006130(-(n+1))*r, for n >= 0, with A052533(-n) = 3*sqrt(-3)^(-n-2)*Snx(-n-2,1/sqrt(-3)), and A006130(-(n+1)) = sqrt(-3)^(-(n+1))*Snx(-(n+1), 1/sqrt(-3)), with the S-Chebyshev polynomials (see A049310), with S(-n, x) = -S(n-2, x), for n>=2, and S(-1, x) = 0. - _Wolfdieter Lang_, Nov 27 2023

%F r = (-1 + sqrt(13))/6 = A223139/3 = 1/A209927.

%e 0.4342585459106648821865368779117493243752160956408743687850755...

%t First[RealDigits[x/.N[Last[Solve[3x^2+x-1==0,x]],78]]] (* _Stefano Spezia_, Aug 29 2022 *)

%Y Cf. A006130, A049310, A052533, A188943, A209927, A223139.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Aug 29 2022