A188640 -id:A188640 - cp's OEIS frontend

A188656 Decimal expansion of (1+sqrt(65))/8.

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

Offset: 1

Clark Kimberling, Apr 09 2011

Apart from the second digit the same as A177707.
Decimal expansion of the shape of a (1/4)-extension rectangle.
See A188640 for definitions of shape and r-extension rectangle.
A (1/4)-extension rectangle matches the continued fraction [1,7,1,1,7,1,1,7,1,1,7,1,1,7,...] for the shape L/W= (1+sqrt(65))/8.  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...].  Specifically, for the (4/3)-extension rectangle, 1 square is removed first, then 7 squares, then 1 square, then 1 square, then 7 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

			length/width = 1.13278221853731870654582665....

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Cf. A188640, A105395.

RealDigits[(1 + Sqrt[65])/8, 10, 111][[1]] (* Robert G. Wilson v, Aug 19 2011 *)

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

A188734 Decimal expansion of (7+sqrt(65))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 12 2011

Apart from the second digit, the same as A171417. - R. J. Mathar, Apr 15 2011
Apart from the first two digits, the same as A188941. - Joerg Arndt, Apr 16 2011
Decimal expansion of the length/width ratio of a (7/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A (7/2)-extension rectangle matches the continued fraction [3,1,3,3,1,3,3,1,3,3,1,3,3,...] for the shape L/W=(7+sqrt(65))/4.  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 (7/2)-extension rectangle, 3 squares are removed first, then 1 square, then 3 squares, then 3 squares,..., so that the original rectangle of shape (7+sqrt(65))/4 is partitioned into an infinite collection of squares.

			3.7655644370746374130916533075759427827835990...

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

Cf. A188640.

SetDefaultRealField(RealField(100)); (7+Sqrt(65))/4; // G. C. Greubel, Nov 01 2018

evalf((7+sqrt(65))/4,140); # Muniru A Asiru, Nov 01 2018

r = 7/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]

default(realprecision, 100); (7+sqrt(65))/4 \\ G. C. Greubel, Nov 01 2018

A188737 Decimal expansion of (7+sqrt(85))/6.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 12 2011

Decimal expansion of the length/width ratio of a (7/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A (7/3)-extension rectangle matches the continued fraction [2,1,2,2,1,2,2,1,2,2,1,...] for the shape L/W=(7+sqrt(85))/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 (7/3)-extension rectangle, 2 squares are removed first, then 1 square, then 2 squares, then 2 squares,..., so that the original rectangle of shape (7+sqrt(85))/6 is partitioned into an infinite collection of squares.

			2.703257409548814551667045713627132192874467508120...

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

Cf. A188640.

SetDefaultRealField(RealField(100)); (7+Sqrt(85))/6; // G. C. Greubel, Nov 01 2018

evalf((7+sqrt(85))/6,140); # Muniru A Asiru, Nov 01 2018

r = 7/3; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

default(realprecision, 100); (7+sqrt(85))/6 \\ G. C. Greubel, Nov 01 2018

A188655 Decimal expansion of (2+sqrt(13))/3.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of the length/width ratio of a (4/3)-extension rectangle.
See A188640 for definitions of shape and r-extension rectangle.
A (4/3)-extension rectangle matches the continued fraction [1,1,6,1,1,1,1,6,1,1,1,1,6,...] for the shape L/W= (2+sqrt(13))/3.  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...].  Specifically, for the (4/3)-extension rectangle, 1 square is removed first, then 1 square, then 6 squares, then 1 square, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.

			length/width = 1.868517091821329764373....

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Cf. A188640, A010122.

r = 4/3; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
RealDigits[(2 + Sqrt@ 13)/3, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[2/3], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

a(130) corrected by Georg Fischer, Apr 01 2020

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

A188727 Decimal expansion of (e+sqrt(16+e^2))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 10 2011

Decimal expansion of shape of an (e/2)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.

An (e/2)-extension rectangle matches the continued fraction A188728 of the shape L/W = (r+sqrt(4+r^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 an (e/2)-extension rectangle, 1 square is removed first, then 1 square, then 7 squares, then 1 square, then 46 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

			1.88862628964821616707581942532177092442419527...

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

Cf. A188640, A188727, A001113.

SetDefaultRealField(RealField(100)); R:= RealField(); (Exp(1) + Sqrt(16 + Exp(2)))/4; // G. C. Greubel, Oct 31 2018

evalf((exp(1)+sqrt(16+exp(2)))/4,140); # Muniru A Asiru, Nov 01 2018

r = e/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A188727 *)
ContinuedFraction[t, 120] (* A188728 *)

default(realprecision, 100); (exp(1) + sqrt(16 + exp(2)))/4 \\ G. C. Greubel, Oct 31 2018

A188728 Continued fraction of (e+sqrt(16+e^2))/4.

Original entry on oeis.org

1, 1, 7, 1, 46, 8, 30, 1, 5, 4, 2, 6, 3, 2, 5, 1, 1, 1, 3, 50, 1, 3, 1, 1, 3, 1, 45, 1, 1, 1, 4, 1, 1, 2, 8, 2, 35, 2, 1, 27, 6, 112, 1, 113, 16, 1, 11, 1, 1, 6, 1, 12, 1, 3, 2, 15, 1, 2, 1, 1, 5, 1, 16, 2, 2, 2, 1, 10, 1, 43, 1, 13, 1, 6, 1, 4, 1, 2, 1, 1, 1, 6, 1, 8, 8, 1, 6, 3, 3, 17, 3, 1, 27, 1, 11, 1, 1, 1, 1, 1, 1, 9, 7, 2, 1, 5, 5, 7, 6, 2, 1, 5, 1, 2, 1, 5, 57, 8, 2, 1

Offset: 0

Clark Kimberling, Apr 10 2011

See A188727 for the origin of the constant.

			(e+sqrt(16+e^2))/4 = [1,1,7,1,46,30,1,5,4,...].

G. C. Greubel, Table of n, a(n) for n = 0..9999

Cf. A188640, A188727 (decimal expansion).

SetDefaultRealField(RealField(100)); ContinuedFraction((Exp(1)  + Sqrt(16 + Exp(2)))/4); // G. C. Greubel, Oct 31 2018

r = e/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A188727 *)
ContinuedFraction[t, 120] (* A188728 *)

default(realprecision, 100); contfrac((exp(1) + sqrt(16 + exp(2)))/4) \\ G. C. Greubel, Oct 31 2018

Offset changed by Andrew Howroyd, Aug 08 2024

A188888 Continued fraction of sqrt(2 + sqrt(3)) or 2*cos(Pi/12).

Original entry on oeis.org

1, 1, 13, 1, 2, 15, 10, 1, 18, 1, 1, 21, 2, 1, 1, 2, 4, 5, 2, 2, 2, 11, 1, 2, 2, 3, 1, 1, 10, 1, 2, 1, 2, 3, 2, 3, 15, 1, 2, 3, 1, 1, 90, 1, 44, 2, 4, 10, 1, 11, 9, 1, 17, 1, 8, 2, 2, 6, 2, 6, 1, 3, 1, 1, 1, 2, 20, 1, 7, 27, 1, 19, 40, 1, 304, 1, 1, 2, 1, 1, 1, 62, 1, 1, 2, 1, 2, 1, 32, 1, 1, 1, 11, 1, 20, 1, 85, 1, 1, 1, 3, 3, 13, 1, 4, 1, 3, 1, 3, 1, 16, 1, 9, 3, 2, 1, 1, 30, 2, 1

Offset: 0

Clark Kimberling, Apr 12 2011

For a geometric interpretation, see A188640 and A188887.

			sqrt(2+sqrt(3)) = [1,1,13,1,2,15,10,1,18,1,1,21,2,1,1,2,4,...].

G. C. Greubel, Table of n, a(n) for n = 0..9999

Cf. A188887 (decimal expansion).

SetDefaultRealField(RealField(100));  ContinuedFraction(Sqrt(2 + Sqrt(3))); // G. C. Greubel, Sep 29 2018

with(numtheory): cfrac(sqrt(2+sqrt(3)),120,'quotients'); # Muniru A Asiru, Sep 30 2018

r = 2^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
ContinuedFraction[Sqrt[2+Sqrt[3]],120] (* Harvey P. Dale, Jul 19 2014 *)

default(realprecision, 100); contfrac(sqrt(2 + sqrt(3))) \\ G. C. Greubel, Sep 29 2018

Name extended by Greg Dresden, Apr 13 2018

Offset changed by Andrew Howroyd, Aug 08 2024

A188930 Decimal expansion of sqrt(5)+sqrt(6).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a sqrt(20)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(20)-extension rectangle matches the continued fraction [4,1,2,5,1,1,4,1,2,24,1,2,...] for the shape L/W=sqrt(5)+sqrt(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 sqrt(20)-extension rectangle, 4 squares are removed first, then 1 square, then 2 squares, then 5 squares,..., so that the original rectangle of shape sqrt(5)+sqrt(6) is partitioned into an infinite collection of squares.

			4.6855577202829677946064577434371676274...

Cf. A188640, A188931.

r = 48^(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[5]+Sqrt[6],10,150][[1]] (* Harvey P. Dale, Nov 06 2014 *)

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A188656 Decimal expansion of (1+sqrt(65))/8.

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

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A188734 Decimal expansion of (7+sqrt(65))/4.

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A188737 Decimal expansion of (7+sqrt(85))/6.

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A188655 Decimal expansion of (2+sqrt(13))/3.

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

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A188727 Decimal expansion of (e+sqrt(16+e^2))/4.

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Magma