A188930 Decimal expansion of sqrt(5)+sqrt(6).

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

Offset: 1

Author

Clark Kimberling, Apr 13 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of the length/width ratio of a sqrt(20)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(20)-extension rectangle matches the continued fraction [4,1,2,5,1,1,4,1,2,24,1,2,...] for the shape L/W=sqrt(5)+sqrt(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(20)-extension rectangle, 4 squares are removed first, then 1 square, then 2 squares, then 5 squares,..., so that the original rectangle of shape sqrt(5)+sqrt(6) is partitioned into an infinite collection of squares.

Examples

			4.6855577202829677946064577434371676274...
		

Crossrefs

Cf. A188640, A188931.

Programs

Mathematica

r = 48^(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[5]+Sqrt[6],10,150][[1]] (* Harvey P. Dale, Nov 06 2014 *)