A190283 Decimal expansion of 1+sqrt(1+sqrt(2)).

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, 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, 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

Offset: 1

Author

Clark Kimberling, May 07 2011

Keywords

• nonn
• cons

Comments

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.

A quartic integer with minimal polynomial x^4 - 4x^3 + 4x^2 - 2. - Charles R Greathouse IV, Feb 09 2017

Examples

			2.553773974030037307344158953063146948165...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A188635, A190283, A190281, A190282.

Programs

Magma

1+Sqrt(1+Sqrt(2)); // G. C. Greubel, Apr 14 2018

Mathematica

r=2^(1/2)
FromContinuedFraction[{2, r, {2, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190284 *)
RealDigits[N[%%, 120]]     (* A190283 *)
N[%%%, 40]
RealDigits[1+Sqrt[1+Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Aug 18 2024 *)

PARI

sqrt(sqrt(2)+1)+1 \\ Charles R Greathouse IV, Feb 09 2017

PARI

polrootsreal(x^4 - 4*x^3 + 4*x^2 - 2)[2] \\ Charles R Greathouse IV, Feb 09 2017