A189966 Decimal expansion of (3+sqrt(33))/4, which has periodic continued fractions [2,5,2,1,2,5,2,1,...] and [3/2, 1, 3/2, 1, ...].

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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (3+sqrt(33))/4. This rectangle can be partitioned into squares in a manner that matches the continued fraction [2,5,2,1,2,5,2,1,2,5,2,1,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [3/2, 1, 3/2, 1, 3/2, ...]. For details, see A188635.

Apart from the first digit, the same as A188939. - R. J. Mathar, May 16 2011

Examples

			2.18614066163450716496265286705473232955506611449...
		

Links

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

Crossrefs

Cf. A188635, A188485.

Programs

Magma

(3+Sqrt(33))/4 // G. C. Greubel, Jan 12 2018

Mathematica

FromContinuedFraction[{3/2, 1, {3/2, 1}}]
ContinuedFraction[%, 25]  (* [2,5,2,1,2,5,2,1,...] *)
RealDigits[N[%%, 120]]  (* A189966 *)
N[%%%, 40]

PARI

(3+sqrt(33))/4 \\ G. C. Greubel, Jan 12 2018