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

1,1

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.

Links

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

Index entries for algebraic numbers, degree 2.

Formula

Minimal polynomial: 3*x^2 - 7*x - 3. - Amiram Eldar, Jun 01 2026

Example

2.703257409548814551667045713627132192874467508120...

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]
(* Alternative: *)
RealDigits[(7+Sqrt[85])/6, 10, 140][[1]] (* Harvey P. Dale, May 12 2026 *)

Prog

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

Crossrefs

Cf. A188640.

Sequence in context: A247373 A021041 A245975 * A200680 A379388 A260129

Adjacent sequences: A188734 A188735 A188736 * A188738 A188739 A188740

Keyword

nonn,cons

Author

Clark Kimberling, Apr 12 2011

Status

approved