A188935 Decimal expansion of (1+sqrt(37))/6.

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

Offset

1,3

Comments

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

A (1/3)-extension rectangle matches the continued fraction [1,5,1,1,5,1,1,5,1,1,5,...] for the shape L/W=(1+sqrt(37))/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 (1/3)-extension rectangle, 1 square is removed first, then 5 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(37))/6 is partitioned into an infinite collection of squares.

Links

Table of n, a(n) for n=1..130.

Index entries for algebraic numbers, degree 2.

Formula

Equals exp(arcsinh(1/6)). - Amiram Eldar, Jul 04 2023

Example

1.1804604217163699481666140408670111770141616824644...

Mathematica

RealDigits[(1 + Sqrt[37])/6, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

Prog

(PARI) (1+sqrt(37))/6 \\ Charles R Greathouse IV, May 18 2026

Crossrefs

Cf. A188640, A188934.

Sequence in context: A073824 A242673 A388813 * A394602 A155528 A096152

Adjacent sequences: A188932 A188933 A188934 * A188936 A188937 A188938

Keyword

nonn,cons

Author

Clark Kimberling, Apr 13 2011

Status

approved