A188655 Decimal expansion of (2+sqrt(13))/3.

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

Offset: 1

Author

Clark Kimberling, Apr 09 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of the length/width ratio of a (4/3)-extension rectangle.

See A188640 for definitions of shape and r-extension rectangle.

A (4/3)-extension rectangle matches the continued fraction [1,1,6,1,1,1,1,6,1,1,1,1,6,...] for the shape L/W= (2+sqrt(13))/3. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (4/3)-extension rectangle, 1 square is removed first, then 1 square, then 6 squares, then 1 square, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.

Examples

			length/width = 1.868517091821329764373....
		

Links

• Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
• Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Crossrefs

Cf. A188640, A010122.

Programs

Mathematica

r = 4/3; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
RealDigits[(2 + Sqrt@ 13)/3, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[2/3], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

Extensions

a(130) corrected by Georg Fischer, Apr 01 2020