A188939 -id:A188939 - cp's OEIS frontend

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

Clark Kimberling, May 05 2011

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

			2.18614066163450716496265286705473232955506611449...

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

Cf. A188635, A188485.

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

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

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

A188938 Decimal expansion of (7-sqrt(33))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 14 2011

Decimal expansion of the shape (= length/width = (7-sqrt(33))/4) of the lesser (7/2)-contraction rectangle.

See A188738 for an introduction to lesser and greater r-contraction rectangles, their shapes, and partitioning these rectangles into a sets of squares in a manner that matches the continued fractions of their shapes.

			0.31385933836549283503734713294526767044...

Cf. A188738, A188939.

r = 7/2; t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

(7-sqrt(33))/4 \\ Charles R Greathouse IV, Apr 25 2016

a(130) corrected by Georg Fischer, Apr 03 2020

cp's OEIS Frontend

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, ...].

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