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

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

1,1

Comments

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.

Links

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

Index entries for algebraic numbers, degree 8.

Formula

Minimal polynomial: 4*x^8 - 24*x^6 + 24*x^4 - 36*x^2 + 9. - Amiram Eldar, May 30 2026

Example

2.271281562422994142313058068759726855455...

Mathematica

FromContinuedFraction[{3^(1/2), 2^(1/2), {3^(1/2), 2^(1/2)}}]
FullSimplify[%]
RealDigits[N[%%, 120]]
N[%%%, 40]

Prog

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

Crossrefs

Cf. A190257 (continued fraction), A188635, A190184, A190185.

Sequence in context: A201315 A199861 A388165 * A013671 A019807 A362029

Adjacent sequences: A190253 A190254 A190255 * A190257 A190258 A190259

Keyword

nonn,cons,changed

Author

Clark Kimberling, May 06 2011

Extensions

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

Status

approved