A188926 Decimal expansion of sqrt((7+sqrt(13))/6).

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

Offset: 1

Author

Clark Kimberling, Apr 13 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of the length/width ratio of a sqrt(1/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(1/3)-extension rectangle matches the continued fraction [1,3,28,1,2,2,42,1,1,1,4,...] for the shape L/W=sqrt((7+sqrt(13))/6). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the sqrt(1/3)-extension rectangle, 1 square is removed first, then 3 squares, then 28 squares, then 1 square,..., so that the original rectangle of shape sqrt((7+sqrt(13))/6) is partitioned into an infinite collection of squares.

Examples

			1.32950813432787924989572324374094447133596...
		

Crossrefs

Cf. A188540, A188927.

Programs

Mathematica

r = 3^(-1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[Sqrt[(7+Sqrt[13])/6],10,140][[1]] (* Harvey P. Dale, Feb 08 2013 *)