A188720 -id:A188720 - cp's OEIS frontend

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

Clark Kimberling, Apr 10 2011

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 0Then 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.

			Length/width = 5.61466856004905343925478283318633736023982...

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.

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

SetDefaultRealField(RealField(100)); Exp(1) + Sqrt(1 + Exp(2)); // G. C. Greubel, Oct 31 2018

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

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

exp(1)+sqrt(1+exp(2)) \\ Charles R Greathouse IV, Jun 16 2011

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

A188722 Decimal expansion of (Pi+sqrt(4+Pi^2))/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of shape of a Pi-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles having shape r.

A Pi-extension rectangle matches the continued fraction A188723 of the shape L/W = (Pi+sqrt(4+Pi^2))/2. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for a Pi-extension rectangle, 3 squares are removed first, then 2 squares, then 3 squares, then 4 squares, then 2 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

			3.4328922159134832442014603702358109669027341058202444195...

Index entries for transcendental numbers

Cf. A188640, A188723, A188720, A000796.

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

(Pi+sqrt(4+Pi^2))/2 \\ Michel Marcus, Apr 01 2015

(Pi+sqrt(4+Pi^2))/2 = [Pi,Pi,Pi,...] (continued fraction). - Clark Kimberling, Sep 23 2013

A188721 Continued fraction of (e+sqrt(4+e^2))/2.

Original entry on oeis.org

3, 21, 2, 40, 1, 8, 1, 18, 1, 4, 2, 7, 14, 25, 1, 2, 1, 4, 1, 1, 1, 1, 2, 8, 50, 4, 1, 1, 3, 1, 11, 1, 1, 2, 3, 1, 1, 3, 1, 2, 22, 1, 1, 4, 1, 4, 1, 1, 4, 4, 2, 2, 2, 57, 1, 1, 34, 5, 1, 2, 2, 1, 1, 8, 13, 2, 3, 3, 17, 1, 1, 49, 1, 2, 1, 5, 1, 7, 1, 9, 1, 11, 1, 1, 7, 13, 1, 1, 59, 4, 8, 1, 3, 1, 4, 6, 1, 9, 11, 1, 1, 4, 456, 2, 8, 23, 2, 4, 2, 2, 1066, 1, 2, 2, 1, 11, 1, 3, 2, 26

Offset: 0

Clark Kimberling, Apr 09 2011

See A188640 and A188720.

Cf. A188640, A188720 (decimal expansion).

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

Offset changed by Andrew Howroyd, Jul 07 2024

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

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A188722 Decimal expansion of (Pi+sqrt(4+Pi^2))/2.

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A188721 Continued fraction of (e+sqrt(4+e^2))/2.

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