A188635 -id:A188635 - cp's OEIS frontend

A190185 Continued fraction of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

1, 1, 5, 1, 6, 1, 5, 1, 1, 40, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 5, 1, 15, 1, 3, 1, 2, 2, 5, 1, 1, 1, 1, 4, 5, 65, 1, 13, 1, 3, 4, 1, 1, 1, 4, 13, 1, 1, 2, 1, 3, 2, 2, 1, 10, 1, 20, 4, 15, 6, 1, 3, 10, 1, 78, 1, 1, 11, 15, 1, 11, 179, 2, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, 2, 6, 1, 1, 7, 5, 1, 4, 1, 9, 1, 1, 2, 10, 3

Offset: 1

Clark Kimberling, May 05 2011

Equivalent to the periodic continued fraction [sqrt(2), sqrt(3), sqrt(2), sqrt(3),...]. For geometric interpretations of both continued fractions, see A190184 and A188635.

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

Cf. A188635, A190184.

ContinuedFraction(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]
ContinuedFraction[Sqrt[1 + Sqrt[2/3] + Sqrt[1 + 2*Sqrt[2/3]]], 100] (* G. C. Greubel, Dec 28 2017 *)

contfrac(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

A190258 Decimal expansion of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2011

The rectangle R whose shape (i.e., length/width) is (x+sqrt(2+4x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 1 in a manner that matches the periodic continued fraction [x, 1, x, 1, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,11,32,1,4,10,2,1,...] at A190259. For details, see A188635.

			2.090657850852244775710089635005221328095...

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

Cf. A190259, A190260, A188635.

[(Sqrt(2) + Sqrt(2+4*Sqrt(2)))/2]; // G. C. Greubel, Dec 26 2017

r=2^(1/2);
FromContinuedFraction[{r,1, {r,1}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190258 *)
RealDigits[N[%%, 120]]     (* A190259 *)
N[%%%, 40]
RealDigits[(Sqrt[2]+Sqrt[2+4Sqrt[2]])/2,10,120][[1]] (* Harvey P. Dale, Jun 20 2021 *)

PARI
```
sqrt(1/2)+sqrt(1/2+sqrt(2))
```

A190259 Continued fraction of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

Original entry on oeis.org

2, 11, 32, 1, 4, 10, 2, 1, 1, 3, 1, 1, 5, 2, 3, 2, 1, 4, 2, 3, 2, 41, 1, 2, 1, 1, 3, 4, 1, 35, 1, 5, 1, 29661, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 5, 2, 2, 2, 1, 1, 1, 5, 15, 2, 1, 1, 1, 2, 7, 1, 1, 1, 13, 1, 1, 1, 1, 20, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 14, 1

Offset: 1

Clark Kimberling, May 06 2011

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

Cf. A188635, A190258.

ContinuedFraction((Sqrt(2) + Sqrt(2+4*Sqrt(2)))/2); // G. C. Greubel, Dec 26 2017

(See A190258.)
ContinuedFraction[(Sqrt[2]+Sqrt[2+4Sqrt[2]])/2,100] (* Harvey P. Dale, Jun 16 2016 *)

contfrac((sqrt(2) + sqrt(2+4*sqrt(2)))/2) \\ G. C. Greubel, Dec 26 2017

Definition clarified by Harvey P. Dale, Jun 16 2016

A190260 Decimal expansion of (1 + sqrt(1 + 2*x))/2, where x=sqrt(2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2011

The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+2x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes 1 and sqrt(2) in a manner that matches the periodic continued fraction [1, x, 1, x, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1, 2,11,32,1,4,10,2,1,...] at A190261. For details, see A188635.

			1.478318343478515956422104436385022215253...

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

Cf. A188635, A190262, A190258.

[(1+Sqrt(1+2*Sqrt(2)))/2]; // G. C. Greubel, Dec 26 2017

r=2^(1/2);
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190261 *)
RealDigits[N[%%, 120]]     (* A190260 *)
N[%%%, 40]

(1+sqrt(1+2*sqrt(2)))/2 \\ G. C. Greubel, Dec 26 2017

A190262 Decimal expansion of (3 + sqrt(9 + 12x))/6, where x=sqrt(3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2011

The rectangle R whose shape (i.e., length/width) is (3+sqrt(9+12x))/6, where x=sqrt(3), can be partitioned into rectangles of shapes 1 and sqrt(3) in a manner that matches the periodic continued fraction [1, x, 1, x, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1, 2, 2, 3, 1, 3, 2, 1, 1, 1, ...] at A190263. For details, see A188635.

			1.409587966713294731518226466119659876240...

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

Cf. A190263, A188635.

[(3 + Sqrt(9 + 12*Sqrt(3)))/6]; // G. C. Greubel, Dec 28 2017

r=3^(1/2)
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190263 *)
RealDigits[N[%%, 120]]     (* A190262 *)
N[%%%, 40]
RealDigits[(3 + Sqrt[9 + 12*Sqrt[3]])/6, 10, 100] (* G. C. Greubel, Dec 28 2017 *)

(3 + sqrt(9 + 12*sqrt(3)))/6 \\ G. C. Greubel, Dec 28 2017

A190281 Decimal expansion of (1+sqrt(1+r))/r, where r=sqrt(2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+r))/r, where r=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 2 in a manner that matches the periodic continued fraction [r, 2, r, 2, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,4,6,1,2,2,2,1,1 ...] at A190282. For details, see A188635.

			1.805790894654357490440645557345527417829...

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

Cf. A190282, A190284.

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

r=2^(1/2)
FromContinuedFraction[{r,2, {r,2}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190282 *)
RealDigits[N[%%, 120]]     (* A190281 *)
N[%%%, 40]
RealDigits[(1 + Sqrt[1 + Sqrt[2]])/Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Jan 31 2018 *)

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

A190283 Decimal expansion of 1+sqrt(1+sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is 1+sqrt(1+sqrt(2)) can be partitioned into rectangles of shapes 2 and sqrt(2) in a manner that matches the periodic continued fraction [2, r, 2, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,1,1,4,6,1,2,2,2,1,1,6,...] at A190284. For details, see A188635.

A quartic integer with minimal polynomial x^4 - 4x^3 + 4x^2 - 2. - Charles R Greathouse IV, Feb 09 2017

			2.553773974030037307344158953063146948165...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4

Cf. A188635, A190283, A190281, A190282.

1+Sqrt(1+Sqrt(2)); // G. C. Greubel, Apr 14 2018

r=2^(1/2)
FromContinuedFraction[{2, r, {2, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190284 *)
RealDigits[N[%%, 120]]     (* A190283 *)
N[%%%, 40]
RealDigits[1+Sqrt[1+Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Aug 18 2024 *)

sqrt(sqrt(2)+1)+1 \\ Charles R Greathouse IV, Feb 09 2017

polrootsreal(x^4 - 4*x^3 + 4*x^2 - 2)[2] \\ Charles R Greathouse IV, Feb 09 2017

A190284 Continued fraction of 1+sqrt(1+sqrt(2)).

Original entry on oeis.org

2, 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, 2, 1, 3, 47, 1, 1, 998, 1

Offset: 1

Clark Kimberling, May 07 2011

1

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

Cf. A188635, A190283, A190282, A154748.

ContinuedFraction(1+Sqrt(1+Sqrt(2))); // G. C. Greubel, Apr 14 2018

FromContinuedFraction[{2, Sqrt[2], {2, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190284 *)
RealDigits[N[%%, 120]]     (* A190283 *)

contfrac(1+sqrt(1+sqrt(2))) \\ G. C. Greubel, Apr 14 2018

A188637 Continued fraction for length/width of a meta-silver rectangle.

Original entry on oeis.org

2, 1, 3, 2, 3, 2, 7, 1, 1, 114, 11, 1, 2, 1, 18, 2, 1, 1, 1, 3, 15, 3, 2, 2, 6, 1, 1, 1, 1, 1, 2, 2, 200, 5, 176, 3, 3, 2, 1, 4, 3, 2, 1, 1, 5, 3, 2, 1, 2, 225, 2, 9, 1, 34, 1, 2, 3, 29, 2, 1, 9, 1, 2, 1, 73, 4, 2, 1, 8, 1, 2, 1, 21, 4, 2, 3, 1, 5, 1, 1, 2, 8, 1, 1, 2, 2, 2, 10, 3, 1, 6, 1, 21, 4, 9, 3, 1, 1, 4, 2, 28, 4, 5, 3, 3, 1, 1, 1, 3, 1, 1, 2, 2, 1, 4, 4, 8, 15, 3

Offset: 0

Clark Kimberling, Apr 06 2011

See A188636 for details.

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

Cf. A188636 (decimal expansion), A188635, A136319.

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

Offset changed by Andrew Howroyd, Jul 08 2024

A189964 Decimal expansion of (3+x+sqrt(38+6*x))/4, where x=sqrt(13).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 04 2011

This constant is the shape of a rectangle whose continued fraction partition matches [r,r,r,...], where r=(3+sqrt(13))/2. For a general discussion, see A188635. The ordinary continued fraction of r is [3,3,3,3,3,3,3,3,3,3,...]. A rectangle of shape r (that is, (length/width)=r) may be compared with the golden rectangle, with shape [1,1,1,1,1,1,...], and the silver rectangle, with shape [2,2,2,2,2,2,...].

			3.5819529507118503770725396959210446869118915483494611612922...

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

Cf. A188635, A188636, A189965.

(3+Sqrt(13)+Sqrt(38+6*Sqrt(13)))/4 // G. C. Greubel, Jan 12 2018

r = (3 +13^(1/2))/2;
FromContinuedFraction[{r, {r}}]
FullSimplify[%]
N[%, 150]
RealDigits[%]  (*A189964*)
ContinuedFraction[%%, 120] (*A189965*)

(3+sqrt(13)+sqrt(38+6*sqrt(13)))/4 \\ G. C. Greubel, Jan 12 2018

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A190185 Continued fraction of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

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A190258 Decimal expansion of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

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A190259 Continued fraction of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

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A190260 Decimal expansion of (1 + sqrt(1 + 2*x))/2, where x=sqrt(2).

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A190262 Decimal expansion of (3 + sqrt(9 + 12x))/6, where x=sqrt(3).

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A190281 Decimal expansion of (1+sqrt(1+r))/r, where r=sqrt(2).

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Magma

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A190283 Decimal expansion of 1+sqrt(1+sqrt(2)).

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