A188640 -id:A188640 - cp's OEIS frontend

A188932 Decimal expansion of sqrt(7)+sqrt(8).

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a sqrt(28)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(28)-extension rectangle matches the continued fraction [5,2,9,5,2,687,6,4,1,2,2,...] for the shape L/W=sqrt(7)+sqrt(8). 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(28)-extension rectangle, 5 squares are removed first, then 2 squares, then 9 squares, then 5 squares,..., so that the original rectangle of shape sqrt(7)+sqrt(8) is partitioned into an infinite collection of squares.

			5.47417843581078068810499320205865658284960293...

Cf. A188640, A188933.

r = 28^(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[7]+Sqrt[8],10,150][[1]] (* Harvey P. Dale, Jun 07 2017 *)

A188935 Decimal expansion of (1+sqrt(37))/6.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a (1/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A (1/3)-extension rectangle matches the continued fraction [1,5,1,1,5,1,1,5,1,1,5,...] for the shape L/W=(1+sqrt(37))/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 (1/3)-extension rectangle, 1 square is removed first, then 5 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(37))/6 is partitioned into an infinite collection of squares.

			1.1804604217163699481666140408670111770141616824644...

Cf. A188640, A188934.

RealDigits[(1 + Sqrt[37])/6, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

Equals exp(arcsinh(1/6)). - Amiram Eldar, Jul 04 2023

A224578 Decimal expansion of (gamma+sqrt(4+gamma^2))/2, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Apr 11 2013

Decimal expansion of shape of a gamma-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.

Specifically, for a gamma-extension rectangle, 1 square is removed first, then 3 squares, then 28 squares, then 13 squares, then 3 squares,...(see A224579), so that the original rectangle is partitioned into an infinite collection of squares.

			1.329422167936173581879417768105... = [gamma, gamma, gamma, ...]

Paolo P. Lava, Table of n, a(n) for n = 1..200
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171

Cf. A001620, A188640, A224579.

SetDefaultRealField(RealField(100)); R:= RealField(); (EulerGamma(R) + Sqrt(4 + EulerGamma(R)^2))/2; // G. C. Greubel, Aug 30 2018

Maple
```
evalf((gamma+sqrt(4+gamma^2))/2,90);
```

RealDigits[(EulerGamma + Sqrt[4 + EulerGamma^2])/2, 10, 100][[1]] (* G. C. Greubel, Aug 30 2018 *)

Euler/2+sqrt(4+Euler^2)/2 \\ Charles R Greathouse IV, Dec 11 2013

A224579 Continued fraction of (gamma+sqrt(4+gamma^2))/2, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

1, 3, 28, 13, 3, 1, 2, 1, 1, 8, 3, 4, 3, 3, 15, 12, 5, 2, 8, 1, 24, 2, 3, 5, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 7, 12, 1, 1, 1, 3, 1, 1, 1, 2, 1, 107, 1, 3, 6, 1, 26, 121, 3, 2, 1, 1, 12, 117, 1, 2, 3, 7, 5, 41, 5, 1, 5, 1, 1, 2, 3, 1, 200, 1, 4, 3, 191, 1, 5, 3, 5

Offset: 0

Paolo P. Lava, Apr 11 2013

Continued fraction of the constant in A224578.

Paolo P. Lava, Table of n, a(n) for n = 0..999

Cf. A001620, A188640, A224578 (decimal expansion).

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction((EulerGamma(R) + Sqrt(4+EulerGamma(R)^2))/2); // G. C. Greubel, Aug 30 2018

Digits:=200; a:=evalf(gamma,5000); evalf((a+sqrt(4+a^2))/2,1000);
numtheory[cfrac](%,200,'quotients') ;

ContinuedFraction[(EulerGamma+Sqrt[4+EulerGamma^2])/2, 100] (* G. C. Greubel, Aug 30 2018 *)

default(realprecision, 100); contfrac((Euler + sqrt(4 + Euler^2))/2) \\ G. C. Greubel, Aug 30 2018

Offset changed by Andrew Howroyd, Aug 08 2024

A320639 Decimal expansion of (C + sqrt(4 + C^2))/2, where C is the Catalan constant.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Oct 18 2018

Decimal expansion of the shape of a Catalan-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.

Specifically, for a Catalan-extension rectangle, 1 square is removed first, then 1 square, then 1 square again, then 3 squares, then 1 square, ... (see A320640), so that the original rectangle is partitioned into an infinite collection of squares.

			1.557868355876025567309823249177... = [Catalan, Catalan, Catalan, ...]

G. C. Greubel, Table of n, a(n) for n = 1..10000
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Cf. A006752, A188640, A320640.

R:= RealField(200); (Catalan(R) + Sqrt(4 + Catalan(R)^2)) / 2; // Vincenzo Librandi, Oct 24 2018

evalf((Catalan+sqrt(4+Catalan^2))/2,135);

First@ RealDigits[(Catalan + Sqrt[4 + Catalan^2])/2, 10, 105] (* Michael De Vlieger, Oct 23 2018 *)

(Catalan+sqrt(4+Catalan^2))/2 \\ Felix Fröhlich, Oct 23 2018

A320640 Continued fraction of (C + sqrt(4 + C^2))/2, where C is the Catalan constant.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 1, 3, 1, 1, 1, 7, 3, 8, 3, 3, 6, 1, 1, 1, 3, 10, 3, 1, 1, 1, 4, 4, 8, 6, 33, 8, 1, 4, 2, 2, 2, 38, 14, 2, 1, 1, 2, 1, 3, 18, 1, 10, 1, 23, 4, 2, 2, 4, 5, 4, 1, 1, 1, 3, 1, 1, 13, 1, 1, 3, 2, 1, 6, 2, 1, 1, 5, 16, 15, 2, 1, 4, 6, 120, 1, 1, 1

Offset: 0

Paolo P. Lava, Oct 18 2018

Continued fraction of the constant in A320639.

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

Cf. A006752, A188640, A320639 (decimal expansion).

SetDefaultRealField(RealField(100)); R:=RealField();  ContinuedFraction((Catalan(R) + Sqrt(4 + Catalan(R)^2))/2); // G. C. Greubel, Oct 31 2018

Digits:=200: evalf((Catalan+sqrt(4+Catalan^2))/2, 1000):
numtheory[cfrac](%, 200,'quotients') ;

ContinuedFraction[(Catalan + Sqrt[4 + Catalan^2])/2, 84] (* Michael De Vlieger, Oct 23 2018 *)

default(realprecision, 100); contfrac((Catalan+sqrt(4+Catalan^2))/2) \\ Michel Marcus, Oct 26 2018

Offset changed by Andrew Howroyd, Aug 10 2024

A366599 Decimal expansion of arcsinh(e).

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, Oct 14 2023

			1.7253825588523150939450...

Cf. A001113, A073742, A085659, A188640, A334399, A365927.

Mathematica
```
RealDigits[ArcSinh[E], 10, 100] [[1]]
```

asinh(exp(1)) \\ Amiram Eldar, Oct 18 2023

Equals log(A188640). - Amiram Eldar, Oct 18 2023

A188723 Continued fraction of (Pi + sqrt(4 + Pi^2))/2.

Original entry on oeis.org

3, 2, 3, 4, 2, 3, 1, 1, 105, 1, 2, 1, 13, 5, 16, 1, 44, 1, 1, 4, 2, 1, 2, 3, 100, 4, 1, 1, 18, 4, 2, 2, 2, 8, 2, 5, 2, 2, 3, 7, 184, 1, 8, 6, 2, 6, 2, 1, 5, 1, 38, 1, 2, 1, 1, 1, 4, 2, 6, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 3, 8, 1, 1, 2, 1, 3, 1, 2, 1, 10, 1, 6, 1, 3, 1, 1, 1, 1, 2, 2, 1, 7, 1, 11, 1, 6, 1, 2, 13, 35, 1, 5, 2, 2, 1, 1, 2, 8, 2, 6, 2, 3, 1, 1, 2, 5

Offset: 0

Clark Kimberling, Apr 09 2011

Continued fraction of the constant in A188722.

Cf. A000796, A188640, A188722 (decimal expansion).

Digits := 100 ;
(Pi+sqrt(4+Pi^2))/2 ;
evalf(%) ;
numtheory[cfrac](%,40,'quotients') ; # R. J. Mathar, Apr 11 2011

r = Pi; 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

A188724 Decimal expansion of shape of a (Pi/2)-extension rectangle; shape = (1/4)*(Pi + sqrt(16 + Pi^2)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

See A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.

A (Pi/2)-extension rectangle matches the continued fraction [2,17,1,1,3,1,3,2,2,1637,1,210,7,...] of the shape L/W = (1/4)*(Pi + sqrt(16 + Pi^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 (Pi/2)-extension rectangle, 2 squares are removed first, then 17 squares, then 1 square, then 1 square, then 3 squares, ..., so that the original rectangle is partitioned into an infinite collection of squares.

			2.0569524387109659093967879243788072585880991...

Index entries for transcendental numbers

Cf. A188640, A188722.

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

a(130) corrected by Georg Fischer, Jul 16 2021

A188731 Decimal expansion of (5+sqrt(41))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 10 2011

Decimal expansion of shape of a (5/2)-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 r.

The continued fractions of the constant are 2, 1, 5, 1, 2, 2, 1, 5, 1, 2, 2, 1, 5, 1, 2, 2, 1, 5, 1...

			2.850781059358212171622054418655453316130105033155254721...

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

Cf. A188640.

SetDefaultRealField(RealField(100)); (5+Sqrt(41))/4; // G. C. Greubel, Nov 01 2018

evalf((5+sqrt(41))/4,140); # Muniru A Asiru, Nov 01 2018

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

default(realprecision, 100); (5+sqrt(41))/4 \\ G. C. Greubel, Nov 01 2018

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A188932 Decimal expansion of sqrt(7)+sqrt(8).

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A188935 Decimal expansion of (1+sqrt(37))/6.

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A224578 Decimal expansion of (gamma+sqrt(4+gamma^2))/2, where gamma is the Euler-Mascheroni constant.

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A224579 Continued fraction of (gamma+sqrt(4+gamma^2))/2, where gamma is the Euler-Mascheroni constant.

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A320639 Decimal expansion of (C + sqrt(4 + C^2))/2, where C is the Catalan constant.

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A320640 Continued fraction of (C + sqrt(4 + C^2))/2, where C is the Catalan constant.

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A366599 Decimal expansion of arcsinh(e).

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