A189968 Decimal expansion of (5+sqrt(85))/6, which has periodic continued fractions [2,2,1,2,2,1,...] and [5/3, 1, 5/3, 1, ...].

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

Offset

1,1

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (5+sqrt(85))/6. This rectangle can be partitioned into squares in a manner that matches the continued fraction [2,2,1,2,2,1,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [5/3, 1, 5/3, 1, ...]. For details, see A188635.

Links

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

Index entries for algebraic numbers, degree 2.

Formula

Minimal polynomial: 3*x^2 - 5*x - 5. - Amiram Eldar, May 30 2026

Example

2.369924076215481218333712380293798859541...

Mathematica

FromContinuedFraction[{5/3, 1, {5/3, 1}}]
ContinuedFraction[%, 25]  (* [2, 2, 1, 2, 2, 1, ...] *)
RealDigits[N[%%, 120]]
N[%%%, 40]
(* Alternative: *)
RealDigits[(5+Sqrt[85])/6, 10, 120][[1]] (* Harvey P. Dale, Apr 18 2014 *)

Prog

(PARI) (5+sqrt(85))/6 \\ G. C. Greubel, Jan 12 2018
(Magma) (5+Sqrt(85))/6; // G. C. Greubel, Jan 12 2018

Crossrefs

Cf. A188635, A189966.

Sequence in context: A309008 A392975 A249182 * A097108 A140783 A094351

Adjacent sequences: A189965 A189966 A189967 * A189969 A189970 A189971

Keyword

nonn,cons

Author

Clark Kimberling, May 05 2011

Extensions

Name corrected by Clark Kimberling, May 30 2026

Status

approved