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A189970 Decimal expansion of (1 + x + sqrt(14+10*x))/4, where x=sqrt(5).

%I A189970 #32 Oct 20 2024 01:41:51
%S A189970 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 A189970 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 A189970 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 A189970 Decimal expansion of (1 + x + sqrt(14+10*x))/4, where x=sqrt(5).
%C A189970 Let R denote a rectangle whose shape (i.e., length/width) is (1 + x + sqrt(14+10*x))/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 A189970 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 A189970 A quartic integer. - _Charles R Greathouse IV_, Aug 29 2015
%H A189970 G. C. Greubel, <a href="/A189970/b189970.txt">Table of n, a(n) for n = 1..10000</a>
%H A189970 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%e A189970 2.31651242917313233045161321161782337624579...
%t A189970 r = (1 + 5^(1/2))/2;
%t A189970 FromContinuedFraction[{r, 1, {r, 1}}]
%t A189970 FullSimplify[%]
%t A189970 ContinuedFraction[%, 100]  (* A189971 *)
%t A189970 RealDigits[N[%%, 120]]     (* A189970 *)
%t A189970 N[%%%, 40]
%t A189970 RealDigits[(1+Sqrt[5]+Sqrt[14+10Sqrt[5]])/4,10,120][[1]] (* _Harvey P. Dale_, Sep 24 2015 *)
%o A189970 (PARI) default(realprecision,1000);x=sqrt(5);(1+x+sqrt(14+10*x))/4 \\ _Anders Hellström_, Aug 24 2015
%o A189970 (PARI) polrootsreal(x^4-x^3-2*x^2-2*x-1)[2] \\ _Charles R Greathouse IV_, Aug 29 2015
%o A189970 (Magma) (1 + Sqrt(5) + Sqrt(14 + 10*Sqrt(5)) )/4; // _G. C. Greubel_, Jan 12 2018
%Y A189970 Cf. A188635, A001622, A189971, A190157.
%K A189970 nonn,cons,easy
%O A189970 1,1
%A A189970 _Clark Kimberling_, May 05 2011