A190179 Decimal expansion of (1+sqrt(-3+4*sqrt(2)))/2.

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, 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, 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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons

Comments

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.

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

Examples

			1.314992983020771197119164203638263044565...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10001
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A188635, A190180, A190177, A190178.

Programs

Magma

(1+Sqrt(-3+4*Sqrt(2)))/2; // G. C. Greubel, Dec 28 2017

Mathematica

r = 1 + 2^(1/2);
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190180 *)
RealDigits[N[%%, 120]]     (* A190179 *)
N[%%%, 40]
RealDigits[(1+Sqrt[4Sqrt[2]-3])/2,10,120][[1]] (* Harvey P. Dale, May 19 2012 *)

PARI

(1+sqrt(-3+4*sqrt(2)))/2 \\ Altug Alkan, Apr 29 2016

Formula

Equals 1+sqrt(c-sqrt(c+sqrt(c-sqrt(c+ ...)))), with c=sqrt(2). - Stanislav Sykora, Apr 29 2016