A188635 -id:A188635 - cp's OEIS frontend

A189963 Decimal expansion of (5+9*sqrt(5))/12.

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

Offset: 1

Clark Kimberling, May 02 2011

The constant at A189963 is the shape of a rectangle whose continued fraction partition consists of 5 golden rectangles. For a general discussion, see A188635.

			2.09371764979150893897354691821512384324713043637531095986983...

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

Cf. A188635, A189961, A189962.

(5+9*Sqrt(5))/12 // G. C. Greubel, Jan 13 2018

r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (*A189963*)
ContinuedFraction[%%]

(5+9*sqrt(5))/12 \\ G. C. Greubel, Jan 13 2018

Continued fraction (as explained at A189959): [r,r,r,r,r], where r=(1+sqrt(5))/2. Ordinary continued fraction, as given by Mathematica program shown below:

[2,10,1,2,29,1,5,2,1,1,2,1,3,5,1,3,3,10,1,2,29,...].

A190178 Continued fraction of (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2.

Original entry on oeis.org

3, 5, 1, 2, 1, 1, 1, 2, 1, 12, 1, 5, 1, 1, 2, 1, 14, 2, 9, 11, 1, 12, 1, 2, 1, 832, 1, 2, 2, 5, 1, 1, 17, 1, 2, 1, 9, 1, 12, 1, 1, 1, 6, 3, 2, 1, 1, 6, 3, 1, 1, 1, 2, 2, 1, 3, 1, 3, 3, 1, 2, 1, 45, 1, 1, 1, 1, 62, 9, 1, 1, 2, 3, 1, 6, 1, 3, 5, 1, 4

Offset: 1

Clark Kimberling, May 05 2011

Equivalent to the periodic continued fraction [r,1,r,1,...] where r=1+sqrt(2), the silver ratio. For geometric interpretations of both continued fractions, see A189977 and A188635.

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

Cf. A188635, A190177, A190180.

ContinuedFraction((1+Sqrt(2)+Sqrt(7+6*Sqrt(2)))/2); // G. C. Greubel, Dec 28 2017

r = 1 + 2^(1/2);
FromContinuedFraction[{r, 1, {r, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190178 *)
RealDigits[N[%%, 120]]     (* A190177 *)
N[%%%, 40]
ContinuedFraction[(1 + Sqrt[2] + Sqrt[7 + 6*Sqrt[2]])/2, 100] (* G. C. Greubel, Dec 28 2017 *)

contfrac((1+sqrt(2)+sqrt(7+6*sqrt(2)))/2) \\ G. C. Greubel, Dec 28 2017

A190282 Continued fraction of (1+sqrt(1+r))/r, where r=sqrt(2).

Original entry on oeis.org

1, 1, 4, 6, 1, 2, 2, 2, 1, 1, 6, 1, 179, 46, 1, 1, 3, 2, 1, 1, 3, 6, 3, 1, 1, 1, 1, 2, 1, 1, 56, 1, 1, 1, 1, 66, 1, 1, 2, 17, 8, 2, 7, 12, 1, 1, 8, 1, 2, 2, 1, 1, 2, 1, 12, 1, 2, 2, 2, 2, 1, 1, 1, 8, 1, 1, 1, 1, 2, 1, 2, 5, 1, 6, 8, 1, 1, 1, 2, 7, 1, 9, 1, 2, 5, 7, 1, 6, 1, 10, 1

Offset: 1

Clark Kimberling, May 07 2011

Equivalent to the periodic continued fraction [r,2,r,2,...] where r=sqrt(2). For geometric interpretations of both continued fractions, see A190281 and A188635.

a(n) = A154748(n+1) for n > 0. - Georg Fischer, Oct 14 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A154748, A188635, A190282.

ContinuedFraction((1 + Sqrt(1 + Sqrt(2)))/Sqrt(2)); // G. C. Greubel, Jan 31 2018

ContinuedFraction[(1 + Sqrt[1 + Sqrt[2]])/Sqrt[2], 50] (* G. C. Greubel, Jan 31 2018 *)

contfrac((1 + sqrt(1 + sqrt(2)))/sqrt(2)) \\ G. C. Greubel, Jan 31 2018

A188636 Decimal expansion of length/width of a metasilver rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 06 2011

A metasilver rectangle is introduced here as a rectangle such that if a silver rectangle is removed from one end, the remaining rectangle is metasilver.  Recall that a rectangle is silver if the removal of 2 squares from one end leaves a rectangle having the same shape s=(length/width) as the original.  This metasilver ratio is given by
s=2.774622899504489263198249637919477554666...;
s=[r,r,r,r...], a periodic continued fraction, r=1+sqrt(2);
s=[2,1,3,2,3,2,7,1,1,114,11,1,2,1,...], as at A188637.

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

Cf. A188637, A136319, A188635, A188638, A188639.

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

Equals (1+sqrt(2)+sqrt(H))/2, where H=7+2*sqrt(2).

A189959 Decimal expansion of (4+5*sqrt(2))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 02 2011

Essentially the same as A020789. - R. J. Mathar, May 16 2011

The constant at A189959 is the shape of a rectangle whose continued fraction partition consists of 3 silver rectangles. For a general discussion, see A188635.

			2.767766952966368811002110905262122598212089844221...

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

Cf. A188635.

(4+5*Sqrt(2))/4 // G. C. Greubel, Jan 13 2018

r=1+2^(1/2);
FromContinuedFraction[{r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]
ContinuedFraction[%%]
RealDigits[(4+5Sqrt[2])/4,10,150][[1]] (* Harvey P. Dale, Dec 17 2024 *)

(4+5*sqrt(2))/4 \\ G. C. Greubel, Jan 13 2018

Continued fraction (as explained at A188635): [r,r,r], where r = 1 + sqrt(2). The ordinary continued fraction (as given by Mathematica program shown below) is as follows:

[2,1,3,3,3,1,2,1,3,3,3,1,2,1,3,3,3,1,2,1,3,3,3,1,2...]

A189961 Decimal expansion of (5+7*sqrt(5))/10.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 02 2011

The constant at A189961 is the shape of a rectangle whose continued fraction partition consists of 3 golden rectangles. For a general discussion, see A188635.

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

Cf. A000032, A000045, A001622, A188635, A189962, A189963.

(5+7*Sqrt(5))/10 // G. C. Greubel, Jan 13 2018

r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (* A189961 *)
ContinuedFraction[%%]
RealDigits[(5+7*Sqrt[5])/10,10,150][[1]] (* Harvey P. Dale, Mar 30 2024 *)

(5+7*sqrt(5))/10 \\ G. C. Greubel, Jan 13 2018

Continued fraction (as explained at A188635): [r,r,r], where r = (1 + sqrt(5))/2. Ordinary continued fraction, as given by Mathematica program shown below:
[2,15,3,15,3,15,3,15,3,...]
From Amiram Eldar, Feb 06 2022: (Start)
Equals phi^4/sqrt(5) - 1, where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+4)/Lucas(k) - 1. (End)

A189962 Decimal expansion of 3(1 + 3sqrt(5))/11.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 02 2011

The constant at A189961 is the shape of a rectangle whose continued fraction partition consists of 4 golden rectangles. For a general discussion, see A188635.

			2.10223743613619156978932391078013510172414229422761...

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

Cf. A188635, A189961, A189963.

3*(1+3*Sqrt(5))/11 // G. C. Greubel, Jan 13 2018

r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (*A189962*)
ContinuedFraction[%%]
RealDigits[3 (1+3*Sqrt[5])/11,10,150][[1]] (* Harvey P. Dale, Sep 11 2023 *)

3*(1+3*sqrt(5))/11 \\ G. C. Greubel, Jan 13 2018

Continued fraction (as explained at A188635): [r,r,r,r], where r = (1 + sqrt(5))/2. Ordinary continued fraction, as given by Mathematica program shown below:

[2,9,1,3,1,1,3,9,1,3,1,1,3,9,1,3,1,1,3,...]

Definition corrected by G. C. Greubel, Jan 13 2018

A189971 Continued fraction of (1 + x + sqrt(14 + 10*x))/4, where x=sqrt(5).

Original entry on oeis.org

2, 3, 6, 3, 1, 2, 15, 2, 3, 6, 1, 7, 1, 4, 2, 3, 1, 4, 2, 1, 1, 1, 2, 1, 20, 17, 3, 1, 2, 3, 1, 1, 3, 1, 4, 9, 73, 1, 37, 192, 3, 1, 1, 1, 1, 5, 1, 21, 1, 6, 7, 1, 3, 3, 1, 8, 2, 2, 1, 1, 8, 1, 2, 1, 1, 8, 1, 2, 1, 20, 2, 16, 3, 19, 2, 1, 3, 7, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 9, 32, 1, 1, 10, 5, 1, 7, 5, 1, 1, 1

Offset: 1

Clark Kimberling, May 05 2011

Equivalent to the periodic continued fraction [r,1,r,1,...] where r=(1+sqrt(5))/2, the golden ratio. For geometric interpretations of both continued fractions, see A189970 and A188635.

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

Cf. A001622, A189970, A188635, A190157, A190158.

ContinuedFraction( (1 + Sqrt(5) + Sqrt(14 + 10*Sqrt(5)) )/4 ); // G. C. Greubel, Jan 12 2018

(See A189970.)
ContinuedFraction[(1+Sqrt[5]+Sqrt[14+10Sqrt[5]])/4,120] (* Harvey P. Dale, Jul 31 2013 *)

contfrac((1+sqrt(5)+sqrt(14+10*sqrt(5)))/4) \\ G. C. Greubel, Jan 12 2018

A190180 Continued fraction of (1+sqrt(-3+4*sqrt(2)))/2.

Original entry on oeis.org

1, 3, 5, 1, 2, 1, 1, 1, 2, 1, 12, 1, 5, 1, 1, 2, 1, 14, 2, 9, 11, 1, 12, 1, 2, 1, 832, 1, 2, 2, 5, 1, 1, 17, 1, 2, 1, 9, 1, 12, 1, 1, 1, 6, 3, 2, 1, 1, 6, 3, 1, 1, 1, 2, 2, 1, 3, 1, 3, 3, 1, 2, 1, 45, 1, 1, 1, 1, 62, 9, 1, 1, 2, 3, 1, 6, 1, 3, 5, 1

Offset: 1

Clark Kimberling, May 05 2011

Equivalent to the periodic continued fraction [1,r,1,r,...] where r=1+sqrt(2), the silver ratio. For geometric interpretations of both continued fractions, see A189979 and A188635.

1 followed by A190178.

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

Cf. A190179, A188635, A190177, A190178.

ContinuedFraction((1+Sqrt(-3+4*Sqrt(2)))/2); // G. C. Greubel, Dec 28 2017

r = 1 + 2^(1/2);
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190180 *)
RealDigits[N[%%, 120]]     (* A190179 *)
N[%%%, 40]
ContinuedFraction[(1 + Sqrt[-3 + 4*Sqrt[2]])/2, 100] (* G. C. Greubel, Dec 28 2017 *)

contfrac((1+sqrt(-3+4*sqrt(2)))/2) \\ G. C. Greubel, Dec 28 2017

A190184 Decimal expansion of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

The rectangle R whose shape (i.e., length/width) is sqrt(1+x+sqrt(1+2x)), where x=sqrt(2/3), can be partitioned into rectangles of shapes sqrt(2) and sqrt(3) in a manner that matches the periodic continued fraction [sqrt(2), sqrt(3), sqrt(2), sqrt(3),...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,5,1,6,1,5,1,1,...] at A190185. For details, see A188635.

			1.854493630042558263683640132452778477778...

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

Cf. A190185.

[Sqrt(1+Sqrt(2/3)+Sqrt(1+2*Sqrt(2/3)))]; // G. C. Greubel, Dec 28 2017

FromContinuedFraction[{2^(1/2), 3^(1/2), {2^(1/2), 3^(1/2)}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190185 *)
RealDigits[N[%%, 120]]      (* A190186 *)
N[%%%, 40]
RealDigits[Sqrt[1+Sqrt[2/3]+Sqrt[1+2*Sqrt[2/3]]], 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

sqrt(1+sqrt(2/3)+sqrt(1+2*sqrt(2/3))) \\ G. C. Greubel, Dec 28 2017

Definition corrected by Bruno Berselli, May 13 2011

cp's OEIS Frontend

A189963 Decimal expansion of (5+9*sqrt(5))/12.

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A190178 Continued fraction of (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2.

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A190282 Continued fraction of (1+sqrt(1+r))/r, where r=sqrt(2).

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A188636 Decimal expansion of length/width of a metasilver rectangle.

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A189959 Decimal expansion of (4+5*sqrt(2))/4.

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A189961 Decimal expansion of (5+7*sqrt(5))/10.

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A189962 Decimal expansion of 3*(1 + 3*sqrt(5))/11.

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A189962 Decimal expansion of 3(1 + 3sqrt(5))/11.