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%I #15 Aug 18 2024 19:11:57
%S 2,5,5,3,7,7,3,9,7,4,0,3,0,0,3,7,3,0,7,3,4,4,1,5,8,9,5,3,0,6,3,1,4,6,
%T 9,4,8,1,6,4,5,8,3,4,9,9,4,1,0,3,0,7,8,3,6,3,3,2,6,7,1,1,4,8,3,3,3,6,
%U 7,5,2,5,6,7,8,8,7,3,3,1,0,2,7,2,7,9,3,7,8,8,6,1,1,7,4,3,6,7,7,4,4,9,2,8,8,3,7,3,3,5,4,3,6,6,6,6,6,6,1,9
%N Decimal expansion of 1+sqrt(1+sqrt(2)).
%C The rectangle R whose shape (i.e., length/width) is 1+sqrt(1+sqrt(2)) can be partitioned into rectangles of shapes 2 and sqrt(2) in a manner that matches the periodic continued fraction [2, r, 2, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,1,1,4,6,1,2,2,2,1,1,6,...] at A190284. For details, see A188635.
%C A quartic integer with minimal polynomial x^4 - 4x^3 + 4x^2 - 2. - _Charles R Greathouse IV_, Feb 09 2017
%H G. C. Greubel, <a href="/A190283/b190283.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%e 2.553773974030037307344158953063146948165...
%t r=2^(1/2)
%t FromContinuedFraction[{2, r, {2, r}}]
%t FullSimplify[%]
%t ContinuedFraction[%, 100] (* A190284 *)
%t RealDigits[N[%%, 120]] (* A190283 *)
%t N[%%%, 40]
%t RealDigits[1+Sqrt[1+Sqrt[2]],10,120][[1]] (* _Harvey P. Dale_, Aug 18 2024 *)
%o (PARI) sqrt(sqrt(2)+1)+1 \\ _Charles R Greathouse IV_, Feb 09 2017
%o (PARI) polrootsreal(x^4 - 4*x^3 + 4*x^2 - 2)[2] \\ _Charles R Greathouse IV_, Feb 09 2017
%o (Magma) 1+Sqrt(1+Sqrt(2)); // _G. C. Greubel_, Apr 14 2018
%Y Cf. A188635, A190283, A190281, A190282.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, May 07 2011
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