A189969 Decimal expansion of (7 + sqrt(133))/6, which has periodic continued fractions [3,11,3,1,3,11,3,1,...] and [7/3, 1, 7/3, 1, ...].

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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (7+sqrt(133))/6. This rectangle can be partitioned into squares in a manner that matches the continued fraction [3,11,3,1,3,11,3,1,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [7/3, 1, 7/3, 1, ...]. For details, see A188635.

Examples

			3.088760432445132648225697206469645416764...
		

Links

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

Crossrefs

Cf. A188635, A188967.

Programs

Magma

(7 + Sqrt(133))/6 // G. C. Greubel, Jan 12 2018

Mathematica

FromContinuedFraction[{7/3, 1, {7/3, 1}}]
ContinuedFraction[%, 25]  (* [3,11,3,1,3,11,3,1,...] *)
RealDigits[N[%%, 120]]  (* A189969 *)
N[%%%, 40]

PARI

(7 + sqrt(133))/6 \\ G. C. Greubel, Jan 12 2018

Extensions

Typo in name corrected by G. C. Greubel, Jan 12 2018