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%I #29 Aug 21 2023 12:14:21
%S 2,3,1,6,5,1,2,4,2,9,1,7,3,1,3,2,3,3,0,4,5,1,6,1,3,2,1,1,6,1,7,8,2,3,
%T 3,7,6,2,4,5,7,9,3,7,3,8,5,8,1,3,8,7,0,8,1,8,9,4,0,6,4,3,0,5,4,4,0,2,
%U 7,5,9,2,1,4,3,8,5,9,8,8,7,1,3,3,7,3,0,9,4,5,7,6,8,2,5,5,4,8,1,5,4,7,2,0,1,4,5,2,5,1,1,1,5,3,5,2,6,9,8,2
%N Decimal expansion of (1 + x + sqrt(14+10*x))/4, where x=sqrt(5).
%C Let R denote a rectangle whose shape (i.e., length/width) is (1 + x + sqrt(14+10x))/4, where x=sqrt(5)). This rectangle can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,r,1,r,1,r,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,3,6,3,...] at A189971. For details, see A188635.
%C Decimal expansion of sqrt(r + r*sqrt(r + r*sqrt(r + ...))), where r = (1 + sqrt(5))/2 = golden ratio. - _Ilya Gutkovskiy_, Aug 24 2015
%C A quartic integer. - _Charles R Greathouse IV_, Aug 29 2015
%H G. C. Greubel, <a href="/A189970/b189970.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 2.31651242917313233045161321161782337624579...
%t r = (1 + 5^(1/2))/2;
%t FromContinuedFraction[{r, 1, {r, 1}}]
%t FullSimplify[%]
%t ContinuedFraction[%, 100] (* A189971 *)
%t RealDigits[N[%%, 120]] (* A189970 *)
%t N[%%%, 40]
%t RealDigits[(1+Sqrt[5]+Sqrt[14+10Sqrt[5]])/4,10,120][[1]] (* _Harvey P. Dale_, Sep 24 2015 *)
%o (PARI) default(realprecision,1000);x=sqrt(5);(1+x+sqrt(14+10*x))/4 \\ _Anders Hellström_, Aug 24 2015
%o (PARI) polrootsreal(x^4-x^3-2*x^2-2*x-1)[2] \\ _Charles R Greathouse IV_, Aug 29 2015
%o (Magma) (1 + Sqrt(5) + Sqrt(14 + 10*Sqrt(5)) )/4; // _G. C. Greubel_, Jan 12 2018
%Y Cf. A188635, A001622, A189971, A190157.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, May 05 2011
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