A188737 Decimal expansion of (7+sqrt(85))/6.

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

Offset: 1

Author

Clark Kimberling, Apr 12 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of the length/width ratio of a (7/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A (7/3)-extension rectangle matches the continued fraction [2,1,2,2,1,2,2,1,2,2,1,...] for the shape L/W=(7+sqrt(85))/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 (7/3)-extension rectangle, 2 squares are removed first, then 1 square, then 2 squares, then 2 squares,..., so that the original rectangle of shape (7+sqrt(85))/6 is partitioned into an infinite collection of squares.

Examples

			2.703257409548814551667045713627132192874467508120...
		

Links

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

Crossrefs

Cf. A188640.

Programs

Magma

SetDefaultRealField(RealField(100)); (7+Sqrt(85))/6; // G. C. Greubel, Nov 01 2018

Maple

evalf((7+sqrt(85))/6,140); # Muniru A Asiru, Nov 01 2018

Mathematica

r = 7/3; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

PARI

default(realprecision, 100); (7+sqrt(85))/6 \\ G. C. Greubel, Nov 01 2018