A188922 Decimal expansion of (sqrt(3) + sqrt(7))/2.

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

Offset

1,1

Comments

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

A sqrt(3)-extension rectangle matches the continued fraction [2,5,3,2,2,9,1,2,1,2,1,9,...] for the shape L/W=(sqrt(3)+sqrt(7))/2. 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(3)-extension rectangle, 2 squares are removed first, then 5 squares, then 3 squares, then 2 squares, ..., so that the original rectangle of shape (sqrt(3)+sqrt(7))/2 is partitioned into an infinite collection of squares.

Links

Table of n, a(n) for n=1..129.

Formula

(sqrt(3)+sqrt(7))/2 = exp(asinh(cos(Pi/6))). - Geoffrey Caveney, Apr 23 2014

cos(Pi/6) + sqrt(1+cos(Pi/6)^2). - Geoffrey Caveney, Apr 23 2014

Example

2.1889010593167339420145310475725663963265322184...

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[3]+Sqrt[7])/2, 10, 140][[1]] (* Harvey P. Dale, Feb 27 2023 *)

Crossrefs

Cf. A188640, A188923.

Sequence in context: A016446 A254794 A086657 * A036296 A078105 A075513

Adjacent sequences: A188919 A188920 A188921 * A188923 A188924 A188925

Keyword

nonn,cons

Author

Clark Kimberling, Apr 13 2011

Status

approved