A190184 -id:A190184 - cp's OEIS frontend

A190185 Continued fraction of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

1, 1, 5, 1, 6, 1, 5, 1, 1, 40, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 5, 1, 15, 1, 3, 1, 2, 2, 5, 1, 1, 1, 1, 4, 5, 65, 1, 13, 1, 3, 4, 1, 1, 1, 4, 13, 1, 1, 2, 1, 3, 2, 2, 1, 10, 1, 20, 4, 15, 6, 1, 3, 10, 1, 78, 1, 1, 11, 15, 1, 11, 179, 2, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, 2, 6, 1, 1, 7, 5, 1, 4, 1, 9, 1, 1, 2, 10, 3

Offset: 1

Clark Kimberling, May 05 2011

Equivalent to the periodic continued fraction [sqrt(2), sqrt(3), sqrt(2), sqrt(3),...]. For geometric interpretations of both continued fractions, see A190184 and A188635.

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

Cf. A188635, A190184.

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

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

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

Definition corrected by Bruno Berselli, May 13 2011

A190256 Decimal expansion of sqrt((3 + x + sqrt(9+6x))/2), where x=sqrt(6).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2011

The rectangle R whose shape (i.e., length/width) is sqrt(3+x+sqrt(9+6x)), where x=sqrt(6), can be partitioned into rectangles of shapes sqrt(3) and sqrt(2) in a manner that matches the periodic continued fraction [sqrt(3), sqrt(2), sqrt(3), sqrt(2), ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,3,1,2,5,2,1,5,95,1,...] at A190257. For details, see A188635.

			2.271281562422994142313058068759726855455...

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

Cf. A190256, A190184, A190185.

[Sqrt((3+Sqrt(6)+Sqrt(9+6*Sqrt(6)))/2)]; // G. C. Greubel, Dec 26 2017

FromContinuedFraction[{3^(1/2), 2^(1/2), {3^(1/2), 2^(1/2)}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190256 *)
RealDigits[N[%%, 120]]     (* A190257 *)
N[%%%, 40]

sqrt((3+sqrt(6)+sqrt(9+6*sqrt(6)))/2) \\ G. C. Greubel, Dec 26 2017

Name corrected by T. D. Noe, Feb 25 2013

cp's OEIS Frontend

A190185 Continued fraction of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

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