A188640 -id:A188640 - cp's OEIS frontend

A176537 Decimal expansion of 5 + sqrt(26).

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

Offset: 2

Klaus Brockhaus, Apr 24 2010

Continued fraction expansion of 5 + sqrt(26) is A010692.

This is the shape of a 10-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011

			5+sqrt(26) = 10.09901951359278483002...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2.

Cf. A010481 (decimal expansion of sqrt(26)), A010692 (all 10's sequence).

r=10; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[5+Sqrt[26],10,120][[1]] (* Harvey P. Dale, Jun 24 2013 *)

5+sqrt(26) \\ Michel Marcus, Jul 23 2018

a(n) = A010481(n-2) for n > 3.
Equals exp(arcsinh(5)), since arcsinh(x) = log(x + sqrt(x^2 + 1)). - Stanislav Sykora, Nov 01 2013
Equals limit_{n->infinity} S(n, 2*sqrt(2*13))/ S(n-1, 2*sqrt(2*13)), with the S-Chebyshev polynomilas (see A049310). - Wolfdieter Lang, Nov 15 2023

A188659 Decimal expansion of (1+sqrt(26))/5.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of the shape of a (2/5)-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, 4, 1, 1, 4, 1, ... = A146325.

			1.219803902718556966005644821804556397912754189219919281516994...

Index entries for algebraic numbers, degree 2.

Cf. A146325, A188640, A188658, A188729, A188730, A010481.

RealDigits[(1 + Sqrt[26])/5, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

Equals exp(arcsinh(1/5)). - Amiram Eldar, Jul 04 2023

A188726 Continued fraction of the shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2).

Original entry on oeis.org

6, 2, 3, 1, 1, 3, 2, 1, 16, 47, 1, 4, 2, 7, 1, 5, 317, 4, 1, 1, 1, 2, 13, 1, 38, 37, 1, 4, 1, 13, 1, 59, 3, 17, 1, 2, 2, 2, 5, 1, 3, 1, 3, 9, 1, 3, 4, 1, 2, 2, 1, 1, 2, 1, 23, 8, 9, 84, 1, 3, 1, 2, 1, 1, 3, 5, 5, 1, 1, 16, 1, 8, 4, 11, 1, 3, 1, 16, 4, 1, 1, 1, 1, 18, 1, 12, 1, 21, 3, 3, 1, 2, 4, 2, 10, 3, 5, 6, 1, 1, 25, 4, 10, 1, 5, 2, 1, 4, 16, 2, 5, 4, 2, 1, 4, 1, 1, 2, 1, 1

Offset: 0

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 2*Pi-extension rectangle matches the continued fraction [6,2,3,1,1,3,2,1,16,47,...] of the shape L/W = Pi + sqrt(1 + 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 (2*Pi)-extension rectangle, 6 squares are removed first, then 2 squares, then 3 squares, then 1 square, then 1 square, ..., so that the original rectangle is partitioned into an infinite collection of squares.

			6.4385009630654083972232325635946917292621665408132...

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

Cf. A188640, A188725 (decimal expansion).

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(Pi(R) + Sqrt(1 + Pi(R)^2)); // G. C. Greubel, Oct 31 2018

with(numtheory): cfrac(Pi+sqrt(1+Pi^2),120,'quotients'); # Muniru A Asiru, Nov 22 2018

r = 2*Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A188725 *)
ContinuedFraction[t, 120] (* A188726 *)
ContinuedFraction[Pi + Sqrt[1 + Pi^2], 100] (* G. C. Greubel, Oct 31 2018 *)

default(realprecision, 100); contfrac(Pi + sqrt(1 + Pi^2)) \\ G. C. Greubel, Oct 31 2018

A089078 Continued fraction for sqrt(2) + sqrt(3).

Original entry on oeis.org

3, 6, 1, 5, 7, 1, 1, 4, 1, 38, 43, 1, 3, 2, 1, 1, 1, 1, 2, 4, 1, 4, 5, 1, 5, 1, 7, 22, 2, 5, 1, 1, 2, 1, 1, 31, 2, 1, 1, 3, 1, 44, 1, 89, 1, 8, 5, 2, 5, 1, 1, 4, 2, 8, 1, 17, 1, 4, 3, 4, 3, 2, 1, 1, 4, 2, 1, 9, 1, 15, 13, 1, 39, 20, 2, 152, 3, 2, 4, 1, 30, 1, 3, 1, 2, 1, 2, 16, 3, 24, 1, 9, 1, 172, 3, 1, 1

Offset: 0

Jeppe Stig Nielsen, Dec 04 2003

This is the most natural example of the fact that the sum of two periodic continued fractions need not have a periodic continued fraction.

a(n) is the numbers of squares removed at stage n of the continued-fraction partitioning of a rectangle of length L and width W satisfying W=L*sqrt(8); see A188640. - Clark Kimberling, Apr 13 2011

G. C. Greubel, Table of n, a(n) for n = 0..5000
G. Xiao, Contfrac

Cf. A135611.

r = 8^(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],100] (* Harvey P. Dale, Aug 17 2019 *)

contfrac(sqrt(2)+sqrt(3)) \\ Michel Marcus, Mar 12 2017

A188658 Decimal expansion of (1+sqrt(101))/10.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 10 2011

Decimal expansion of the shape of a (1/5)-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 is 1, 9, 1, 1, 9, 1, 1, 9, 1, 1, 9, 1, 1, 9, 1...

			1.104987562112089027021926491275957618694502347002...

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

Cf. A188640.

r = 1/5; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[(1+Sqrt[101])/10,10,150][[1]] (* Harvey P. Dale, Nov 29 2020 *)

Equals exp(arcsinh(1/10)). - Amiram Eldar, Jul 04 2023

a(130) corrected by Georg Fischer, Apr 02 2020

A188720 Decimal expansion of (e+sqrt(4+e^2))/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of shape of an e-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.
An e-extension rectangle matches the continued fraction A188721 of the shape L/W = (1/2) *(e+sqrt(4+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 an e-extension rectangle, 3 squares are removed first, then 21 squares, then 2 squares, then 40 squares, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.
(e+sqrt(4+e^2))/2 = [e,e,e,... ] (continued fraction). - Clark Kimberling, Sep 23 2013

			3.046524695333472471811401766587155243274607058879794774577422496312...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A188640, A188721, A188726, A188722, A188725.

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

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

r=E; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(E+Sqrt[4+E^2])/2,10,150][[1]] (* Harvey P. Dale, Jan 07 2015 *)

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

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

A188725 Decimal expansion of shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2).

Original entry on oeis.org

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

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 2*Pi-extension rectangle matches the continued fraction [6,2,3,1,1,3,2,1,16,47,...] of the shape L/W = Pi + sqrt(1 + 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 (2*Pi)-extension rectangle, 6 squares are removed first, then 2 squares, then 3 squares, then 1 square, then 1 square, ..., so that the original rectangle is partitioned into an infinite collection of squares.

			6.4385009630654083972232325635946917292621665408132615256106...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A188640, A188726.

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

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

r = 2*Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A188725 *)
ContinuedFraction[t, 120] (* A188726 *)
RealDigits[Pi + Sqrt[1 + Pi^2], 10, 100][[1]] (* G. C. Greubel, Oct 31 2018 *)

default(realprecision, 100); Pi + sqrt(1 + Pi^2) \\ G. C. Greubel, Oct 31 2018

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

A188934 Decimal expansion of (1+sqrt(17))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a (1/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A (1/2)-extension rectangle matches the continued fraction [1,3,1,1,3,1,1,3,1,1,3,...] for the shape L/W=(1+sqrt(17))/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 (1/2)-extension rectangle, 1 square is removed first, then 3 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(17))/4 is partitioned into an infinite collection of squares.
Conjecture: This number is an eigenvalue to infinitely many n*n submatrices of A191898, starting in the upper left corner, divided by the row index. For the first few characteristic polynomials see A260237 and A260238. - Mats Granvik, May 12 2016.

			1.2807764064044151374553524639935192562...

Cf. A188640, A188935.

Essentially the same as A188485.

r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
(* for the continued fraction *) ContinuedFraction[t, 120]
RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[1/4], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

(sqrt(17)+1)/4 \\ Charles R Greathouse IV, May 12 2016

cp's OEIS Frontend

A176537 Decimal expansion of 5 + sqrt(26).

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A188659 Decimal expansion of (1+sqrt(26))/5.

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A188726 Continued fraction of the shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2).

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A089078 Continued fraction for sqrt(2) + sqrt(3).

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A188658 Decimal expansion of (1+sqrt(101))/10.

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A188720 Decimal expansion of (e+sqrt(4+e^2))/2.

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

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