A190179 - OEIS

%I #26 Dec 10 2024 05:38:46

%S 1,3,1,4,9,9,2,9,8,3,0,2,0,7,7,1,1,9,7,1,1,9,1,6,4,2,0,3,6,3,8,2,6,3,

%T 0,4,4,5,6,4,9,0,9,3,4,6,6,3,3,7,5,6,0,0,3,2,0,8,0,0,3,1,7,2,6,0,5,6,

%U 0,2,8,8,6,5,3,6,0,3,8,8,6,6,1,9,2,6,2,4,0,6,2,5,8,0,8,8,0,9,3,2,4,8,0,9,9,1,8,4,8,1,5,5,0,8,9,5,5,3,9,1

%N Decimal expansion of (1+sqrt(-3+4*sqrt(2)))/2.

%C Let R denote a rectangle whose shape (i.e., length/width) is (1+sqrt(-3+4*sqrt(2)))/2.  R can be partitioned into squares and silver rectangles in a manner that matches the periodic continued fraction [1,r,1,r,...], where r is the silver ratio: 1+sqrt(2)=[2,2,2,2,2,...].  R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,3,5,1,2,1,1,1,2,...] at A190180.  For details, see A188635.

%C The real value a-1 is the only invariant point of the complex-plane mapping M(c,z)=sqrt(c-sqrt(c+z)), with c = sqrt(2), and its only attractor, convergent from any starting complex-plane location. - _Stanislav Sykora_, Apr 29 2016

%H Chai Wah Wu, <a href="/A190179/b190179.txt">Table of n, a(n) for n = 1..10001</a>

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%F Equals 1+sqrt(c-sqrt(c+sqrt(c-sqrt(c+ ...)))), with c=sqrt(2). - _Stanislav Sykora_, Apr 29 2016

%e 1.314992983020771197119164203638263044565...

%t r = 1 + 2^(1/2));

%t FromContinuedFraction[{1, r, {1, r}}]

%t FullSimplify[%]

%t ContinuedFraction[%, 100]  (* A190180 *)

%t RealDigits[N[%%, 120]]     (* A190179 *)

%t N[%%%, 40]

%t RealDigits[(1+Sqrt[4Sqrt[2]-3])/2,10,120][[1]] (* _Harvey P. Dale_, May 19 2012 *)

%o (PARI) (1+sqrt(-3+4*sqrt(2)))/2 \\ _Altug Alkan_, Apr 29 2016

%o (Magma) (1+Sqrt(-3+4*Sqrt(2)))/2; // _G. C. Greubel_, Dec 28 2017

%Y Cf. A188635, A190180, A190177, A190178.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, May 05 2011