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%I #25 Mar 21 2024 08:35:12
%S 3,5,8,2,5,7,5,6,9,4,9,5,5,8,4,0,0,0,6,5,8,8,0,4,7,1,9,3,7,2,8,0,0,8,
%T 4,8,8,9,8,4,4,5,6,5,7,6,7,6,7,9,7,1,9,0,2,6,0,7,2,4,2,1,2,3,9,0,6,8,
%U 6,8,4,2,5,5,4,7,7,7,0,8,8,6,6,0,4,3,6,1,5,5,9,4,9,3,4,4,5,0,3
%N Decimal expansion of (-1 + sqrt(21))/10 = 1/A222134.
%C Positive root of the minimal polynomial x^2 + 1/5 - 1/5. The negative root is -(1/5)*A222134 = -0.558257569...
%C c^n = A(-n) + B(-n)*phi21, and A(n) = S21(n+1) - S21(n) = A365824(n), with phi21 = A222134, and B(n) = S21(n) = A015440(n-1), where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).
%C The formula for negative indices of S is S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.
%H Paolo Xausa, <a href="/A367453/b367453.txt">Table of n, a(n) for n = 0..10000</a>
%F c = 1/phi21 = (1/5)*(1 - phi21), with phi21 = (1 + sqrt(21))/2 = A222134, hence an algebraic number of the real quadratic number field Q(sqrt(21)) but not an algebraic integer like phi24.
%F Equals (A010477-1)/10. - _R. J. Mathar_, Nov 21 2023
%F Equals 2*A222135/10. - _Hugo Pfoertner_, Mar 21 2024
%e c = 0.3582575694955840006588047193728008488984456...
%t First[RealDigits[(Sqrt[21]-1)/10,10,100]] (* _Paolo Xausa_, Nov 21 2023 *)
%o (PARI) \\ Works in v2.13 and higher; n = 100 decimal places
%o my(n=100); digits(floor(10^(n-1)*(quadgen(84)-1))) \\ _Michal Paulovic_, Nov 20 2023
%Y Cf. A010477, A049310, A222134, A222135, A365824.
%K nonn,cons,easy
%O 0,1
%A _Wolfdieter Lang_, Nov 20 2023