A189966 - OEIS

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

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OFFSET

1,1

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

LINKS

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

EXAMPLE

2.18614066163450716496265286705473232955506611449...

MATHEMATICA

FromContinuedFraction[{3/2, 1, {3/2, 1}}]

ContinuedFraction[%, 25] (* [2, 5, 2, 1, 2, 5, 2, 1, ...] *)

RealDigits[N[%%, 120]] (* A189966 *)

N[%%%, 40]

PROG

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