A190184 Decimal expansion of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons

Comments

The rectangle R whose shape (i.e., length/width) is sqrt(1+x+sqrt(1+2x)), where x=sqrt(2/3), can be partitioned into rectangles of shapes sqrt(2) and sqrt(3) in a manner that matches the periodic continued fraction [sqrt(2), sqrt(3), sqrt(2), sqrt(3),...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,5,1,6,1,5,1,1,...] at A190185. For details, see A188635.

Examples

			1.854493630042558263683640132452778477778...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A190185.

Programs

Magma

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

Mathematica

FromContinuedFraction[{2^(1/2), 3^(1/2), {2^(1/2), 3^(1/2)}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190185 *)
RealDigits[N[%%, 120]]      (* A190186 *)
N[%%%, 40]
RealDigits[Sqrt[1+Sqrt[2/3]+Sqrt[1+2*Sqrt[2/3]]], 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

PARI

sqrt(1+sqrt(2/3)+sqrt(1+2*sqrt(2/3))) \\ G. C. Greubel, Dec 28 2017

Extensions

Definition corrected by Bruno Berselli, May 13 2011