A188659 -id:A188659 - 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

A146325 Period 3: repeat [1, 4, 1].

Original entry on oeis.org

1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1

Offset: 1

Artur Jasinski, Oct 30 2008

Continued fraction of (1 + sqrt(26))/5 = A188659.
Digital roots of the centered triangular numbers A005448. - Ant King, May 08 2012
Also the digital roots of centered 12-gonal numbers A003154. - Peter M. Chema, Dec 20 2023

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A003154, A005448, A021337, A131534 (square roots), A188659.

&cat [[1,4,1]^^40]; // Bruno Berselli, Jun 27 2016

seq(op([1, 4, 1]), n=1..50); # Wesley Ivan Hurt, Jul 01 2016

Table[Round[N[4 (Cos[(2 n - 1) ArcTan[Sqrt[3]]])^2, 100]], {n, 1, 100}]
PadLeft[{},111,{1,4,1}] (* Harvey P. Dale, Sep 18 2011 *)

a(n)=1+3*(n%3==2) \\ Jaume Oliver Lafont, Mar 24 2009

a(n) = 4*(cos((2*n - 1)*Pi/3))^2 = 4 - 4*(sin((2*n - 1)*Pi/3))^2.
a(n+3) = a(n).
a(n) = 2 - cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3).
O.g.f.: x*(1+4*x+x^2)/(1-x^3). [Richard Choulet, Nov 03 2008]
a(n) = 6 - a(n-1) - a(n-2) for n>2. - Ant King, Jun 12 2012
a(n) = (n mod 3)^(n mod 3). - Bruno Berselli, Jun 27 2016
a(n) = 1 + A021337(n) for n>0. - Wesley Ivan Hurt, Jul 01 2016

A188730 Decimal expansion of (2+sqrt(29))/5.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 10 2011

Decimal expansion of shape of a (4/5)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, shape=length/width, and an r-extension rectangle is composed of two rectangles of shape 1/r when r<1.

The continued fractions of the constant are 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2...

			1.4770329614269008062501420983080659112590240323289577675360...

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

Cf. A188640, A188658, A188659, A188729, A188730.

SetDefaultRealField(RealField(100)); (2+Sqrt(29))/5; // G. C. Greubel, Nov 01 2018

evalf((2+sqrt(29))/5,140); # Muniru A Asiru, Nov 01 2018

RealDigits[(2 + Sqrt[29])/5, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

default(realprecision, 100); (2+sqrt(29))/5 \\ G. C. Greubel, Nov 01 2018

A188729 Decimal expansion of (3+sqrt(109))/10.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 10 2011

Decimal expansion of shape of a (3/5)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.
Briefly, shape=length/width, and an r-extension rectangle is composed of two rectangles of shape 1/r when r<1.
The continued fraction of the constant is 1, 2, 1, 9, 1, 2, 1, 1, 2, 1, 9, 1, 2, 1, 1, 2, 1, 9, 1, 2, 1,...

			1.3440306508910550179757754022548047678289849837719799753005...

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

Cf. A188640, A188658, A188659, A188730.

SetDefaultRealField(RealField(100)); (3+Sqrt(109))/10; // G. C. Greubel, Nov 01 2018

evalf((3+sqrt(109))/10,140); # Muniru A Asiru, Nov 01 2018

r = 3/5; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

default(realprecision, 100); (3+sqrt(109))/10 \\ G. C. Greubel, Nov 01 2018

A344426 Decimal expansion of sqrt(26)/5.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 18 2021

sqrt(26)/5 is the length of the shortest line segment needed to dissect the unit square into 10 regions with equal areas if all the line segments start at the same vertex of the square.

Essentially the same as A188659. - R. J. Mathar, Jun 04 2021

			1.01980390271855696600564482180...

Cf. A344382, A188659.

Mathematica
```
RealDigits[Sqrt[26]/5, 10, 200][[1]]
```

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

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A146325 Period 3: repeat [1, 4, 1].

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A188730 Decimal expansion of (2+sqrt(29))/5.

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A188729 Decimal expansion of (3+sqrt(109))/10.

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A344426 Decimal expansion of sqrt(26)/5.

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