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%I #24 Jan 15 2024 16:44:12
%S 3,9,0,3,8,8,2,0,3,2,0,2,2,0,7,5,6,8,7,2,7,6,7,6,2,3,1,9,9,6,7,5,9,6,
%T 2,8,1,4,3,3,9,9,9,0,3,1,7,1,7,0,2,5,5,4,2,9,9,8,2,9,1,9,6,6,3,6,8,6,
%U 9,2,9,3,2,9,2,2
%N Decimal expansion of the positive root of 4*x^2 + x - 1.
%C The negative root is -(A189038 - 1) = -0.6403882032... .
%C c^n = A052923(-n) + A006131(-(n+1))*phi17, for n >= 0, with phi17 = A222132 = (1 + sqrt(17))/2, A052923(-n) = -(-2*i)^(-n)*S(-(n+2), i/2) = (i/2)^n*S(n, i/2), with i = sqrt(-1), and A006131(-(n+1)) = A052923(-n+1)/4 = -(i/2)^(n+1)*S(n-1, i/2), with the S-Chebyshev polynomials (see A049310), and S(-n, x) = -S(n-2, x), for n >= 1. - _Wolfdieter Lang_, Jan 04 2024
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F c = (-1 + sqrt(17))/8 = A189038 - 5/4 = A174930 - 5/8.
%F c = 1/phi17 = (-1 + phi17)/4, with phi17 = A222132. - _Wolfdieter Lang_, Jan 05 2024
%e c = 0.39038820320220756872767623199675962814339990317170255429982919663...
%t RealDigits[(Sqrt[17] - 1)/8, 10, 120][[1]] (* _Amiram Eldar_, Jan 20 2023 *)
%t RealDigits[Root[4x^2+x-1,2],10,120][[1]] (* _Harvey P. Dale_, Jan 15 2024 *)
%Y Cf. A006131, A010473, A049310, A052923, A174930, A189038, A222132.
%K nonn,cons,easy
%O 0,1
%A _Wolfdieter Lang_, Jan 20 2023
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