A188640 Decimal expansion of e + sqrt(1+e^2).

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

Offset

1,1

Comments

The shape of a rectangle WXYZ, denoted by [WXYZ], is defined by length/width: [WXYZ]=max{|WX|/|YZ|, |YZ|/|WX|}. Consider the following configuration of rectangles AEFD, EBCF, ABCD, where AEFD is not a square:

D................F....C

.......................

.......................

.......................

A................E....B

Let r=[AEFD]. The r-extension rectangle of AEFD is here introduced as the rectangle ABCD for which [AEFD]=[EBCF] and |AE|<>|EB|. That is, AEFD has the prescribed shape r, and AEFD and EBCF are similar without being congruent.

We extend the definition of r-extension rectangle to the case that 0<r<1; in this case, [AEFD]=1/r and ABCD is defined as above.

Then for all r>0, it is easy to prove that [ABCD] = (r+sqrt(4+r^2))/2.

This here is the length/width ratio for the (2e)-extension rectangle.

A (2e)-extension rectangle matches the continued fraction A188796 for the shape L/W=(e+sqrt(1+e^2). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (2e)-extension rectangle, 5 squares are removed first, then 1 square, then 1 square, then 1 square, then 1 square, then 2 squares..., so that the original rectangle is partitioned into an infinite collection of squares.

Shapes of other r-extension rectangles, partitionable into a collection of squares in accord with the continued fraction of the shape [ABCD], are approximated at A188635-A188639, A188655-A188659, and A188720-A188737.

For (related) r-contraction rectangles, see A188738 and A188739.

Links

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

Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Index entries for transcendental numbers.

Formula

Equals exp(A366599). - Amiram Eldar, Oct 18 2023

Example

Length/width = 5.61466856004905343925478283318633736023982...

Maple

evalf(exp(1)+sqrt(1+exp(2)), 140); # Muniru A Asiru, Nov 01 2018

Mathematica

r=2E; t=(r+(4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]

Prog

(PARI) exp(1)+sqrt(1+exp(2)) \\ Charles R Greathouse IV, Jun 16 2011
(Magma) SetDefaultRealField(RealField(100)); Exp(1) + Sqrt(1 + Exp(2)); // G. C. Greubel, Oct 31 2018

Crossrefs

Cf. A001113, A188796, A188738, A188739, A188720, A366599.

Sequence in context: A046614 A371048 A080130 * A198419 A323738 A222133

Adjacent sequences: A188637 A188638 A188639 * A188641 A188642 A188643

Keyword

nonn,cons,easy

Author

Clark Kimberling, Apr 10 2011

Status

approved