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%I #49 Dec 21 2023 03:59:47
%S 2,5,6,1,5,5,2,8,1,2,8,0,8,8,3,0,2,7,4,9,1,0,7,0,4,9,2,7,9,8,7,0,3,8,
%T 5,1,2,5,7,3,5,9,9,6,1,2,6,8,6,8,1,0,2,1,7,1,9,9,3,1,6,7,8,6,5,4,7,4,
%U 7,7,1,7,3,1,6,8,8,1,0,7,9,6,7,9,3,9,3,1,8,2,5,4,0,5,3,4,2,1,4,8,3,4,2,2,7
%N Decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))).
%C Sequence with a(1) = 1 is decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))) = A222133.
%C Because 17 == 1 (mod 4), the basis for integers in the real quadratic number field K(sqrt(17)) is <1, omega(17)>, where omega(17) = (1 + sqrt(17))/2. - _Wolfdieter Lang_, Feb 10 2020
%C This is the positive root of the polynomial x^2 - x - 4, with negative root -A222133. - _Wolfdieter Lang_, Dec 10 2022
%C It is the spectral radius of the diamond graph (see Seeger and Sossa, 2023). - _Stefano Spezia_, Sep 19 2023
%C c^n = A006131(n) + A006131(n-1) * d, where c = (1 + sqrt(17))/2 and d = (-1 + sqrt(17))/2. - _Gary W. Adamson_, Nov 25 2023
%C c^n = A052923(n) + A006131(n-1) * c. Also for negative n. - _Wolfdieter Lang_, Nov 27 2023
%H Alberto Seeger and David Sossa, <a href="https://ajc.maths.uq.edu.au/pdf/87/ajc_v87_p258.pdf">Infinite families of connected graphs with equal spectral radius</a>, Australas. J. Combin. 87 (2) (2023), 258-276. See pp. 260 and 263.
%F Closed form: (sqrt(17) + 1)/2 = A178255 - 1 = A082486 - 2.
%F sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))) - 1 = sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))). See A222133.
%e 2.561552812808830274910704...
%p Digits:=140:
%p evalf((sqrt(17)+1)/2); # _Alois P. Heinz_, Sep 19 2023
%t RealDigits[(1 + Sqrt[17])/2, 10, 130]
%Y Cf. A222133, A178255, A082486.
%Y Cf. A006131, A052923.
%K nonn,cons,easy
%O 1,1
%A _Jaroslav Krizek_, Feb 08 2013
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