A188635 -id:A188635 - cp's OEIS frontend

A190289 Decimal expansion of (3+sqrt(21))/4.

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (3+sqrt(21))/4, can be partitioned into rectangles of shapes 3/2 and 2 in a manner that matches the periodic continued fraction [3/2, 2, 3/2, 2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [1,1,8,1,1,2,1,1,8,1,1,2,,...]. For details, see A188635.

			1.895643923738960001647011798432002122246...

A188635.

FromContinuedFraction[{3/2, 2, {3/2, 2}}]
ContinuedFraction[%, 100]  (* [1,1,8,1,1,2,... *)
RealDigits[N[%%, 120]]     (* A190289 *)
N[%%%, 40]
RealDigits[(3+Sqrt[21])/4,10,120][[1]] (* Harvey P. Dale, Dec 13 2019 *)

A190290 Decimal expansion of (3+sqrt(21))/3.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (3+sqrt(21))/3, can be partitioned into rectangles of shapes 3/2 and 2 in a manner that matches the periodic continued fraction [2, 3/2, 2, 3/2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [2,1,1,8,1,1,2,1,1,8,1,1,2,,...]. For details, see A188635.

			2.527525231651946668862682397909336162995...

Cf. A188635, A190289.

FromContinuedFraction[{2, 3/2, {2, 3/2}}]
ContinuedFraction[%, 100]  (* [2,1,1,8,1,1,2,... *)
RealDigits[N[%%, 120]]     (* A190290 *)
N[%%%, 40]

Equals 1 + Sum_{k>=0} binomial(2*k,k)/7^k. - Amiram Eldar, Aug 03 2020

A189960 Decimal expansion of (9+27*sqrt(2))/17.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 02 2011

The constant at A189960 is the shape of a rectangle whose continued fraction partition consists of 4 silver rectangles. For a general discussion, see A188635.

			2.7755156578866803716262115031565793012577141550...

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

Cf. A188635, A189959.

(9+27*Sqrt(2))/17 // G. C. Greubel, Jan 13 2018

r = 1 + 2^(1/2);
FromContinuedFraction[{r,r,r,r}]
FullSimplify[%]
N[%, 150]
RealDigits[%]  (*A189960*)
ContinuedFraction[%%, 120]
RealDigits[(9+27Sqrt[2])/17,10,150][[1]] (* Harvey P. Dale, Dec 22 2019 *)

(9+27*sqrt(2))/17 \\ G. C. Greubel, Jan 13 2018

Continued fraction (as explained at A188635): [r,r,r,r], where r = 1 + sqrt(2). The ordinary continued fraction (as given by Mathematica program shown below) is as follows: [2,1,3,2,5,76,5,2,3,1,3,1,2,1,1,7,1,10,38,10,...]

A189969 Decimal expansion of (7 + sqrt(133))/6, which has periodic continued fractions [3,11,3,1,3,11,3,1,...] and [7/3, 1, 7/3, 1, ...].

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

Let R denote a rectangle whose shape (i.e., length/width) is (7+sqrt(133))/6. This rectangle can be partitioned into squares in a manner that matches the continued fraction [3,11,3,1,3,11,3,1,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [7/3, 1, 7/3, 1, ...]. For details, see A188635.

			3.088760432445132648225697206469645416764...

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

Cf. A188635, A188967.

(7 + Sqrt(133))/6 // G. C. Greubel, Jan 12 2018

FromContinuedFraction[{7/3, 1, {7/3, 1}}]
ContinuedFraction[%, 25]  (* [3,11,3,1,3,11,3,1,...] *)
RealDigits[N[%%, 120]]  (* A189969 *)
N[%%%, 40]

(7 + sqrt(133))/6 \\ G. C. Greubel, Jan 12 2018

Typo in name corrected by G. C. Greubel, Jan 12 2018

A190181 Decimal expansion of (15+sqrt(465))/12.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

The rectangle R whose shape (i.e., length/width) is (15+sqrt(465))/12 can be partitioned into rectangles of shapes 5/2 and 3/2 in a manner that matches the periodic continued fraction [5/2, 3/2, 5/2, 3/2,...]. R can also be partitioned into squares so as to match the periodic continued fraction [3,21,3,1,1,4,1,4,1,1,3,21,...]. For details, see A188635.

			3.046988221070652056227828483250098729807...

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

Cf. A188635.

[(15+sqrt(465))/12]; // G. C. Greubel, Dec 28 2017

FromContinuedFraction[{5/2, 3/2, {5/2, 3/2}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* [3,21,3,1,1,4,1,4,1,1,3,21,...] *)
RealDigits[N[%%, 120]]      (* A190181 *)
N[%%%, 40]
RealDigits[(15+Sqrt[465])/12, 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

(15+sqrt(465))/12 \\ G. C. Greubel, Dec 28 2017

A190264 Decimal expansion of (sqrt(89) - 6)/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (-6 + sqrt(89))/2 can be partitioned into rectangles of shapes 3/2 and 3 in a manner that matches the periodic continued fraction [3/2, 3, 3/2, 3, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [1, 1, 2, 1, 1, 6, 1, 36, 1, 6, 1, 1, 2, 1, 8, 1, 2, 1, 1, 6, 1, 36, ...]. For details, see A188635.

Quadratic number with denominator 2 and minimal polynomial 4x^2 + 24x - 53. - Charles R Greathouse IV, Apr 21 2016

			1.716990566028301905660330188811320358491...

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

Cf. A188635, A190290, A178255.

FromContinuedFraction[{3/2, 3, {3/2, 3}}]
ContinuedFraction[%, 100]  (* [1, 1, 2, 1, 1, 6, 1, 36, ... *)
RealDigits[N[%%, 120]]     (* A190264 *)
N[%%%, 40]

sqrt(89)/2-3 \\ Charles R Greathouse IV, Apr 21 2016

A190286 Continued fraction of (3+sqrt(9+4r))/2, where r=sqrt(3).

Original entry on oeis.org

3, 2, 55, 6, 1, 1, 1, 9, 1, 1, 1, 7, 2, 2, 1, 4, 2, 6, 1, 27, 1, 10, 5, 1, 1, 2, 2, 3, 6, 6, 1, 9, 1, 1, 5, 1, 2, 3, 94, 1, 13, 18, 7, 1, 1, 1, 4, 1, 20, 1, 2, 7, 11, 1, 26251, 1, 43, 1, 1, 5, 1, 1, 1, 1, 5, 1, 47, 1, 1, 2, 12, 6, 3, 4, 6, 7, 5, 1, 1, 1, 1, 1, 4, 6, 23, 3, 2, 1, 1, 2, 53

Offset: 0

Clark Kimberling, May 07 2011

Equivalent to the periodic continued fraction [3, r, 3, r, ...] where r=sqrt(3). For geometric interpretations of both continued fractions, see A190285 and A188635.

Cf. A188635, A190285.

Mathematica
```
(See A190285.)
```

Offset changed by Andrew Howroyd, Jul 03 2024

A190288 Continued fraction of (5+sqrt(25+4r))/2, where r=sqrt(5).

Original entry on oeis.org

5, 2, 2, 2, 1, 1, 1, 10, 1, 1, 2, 1, 2, 1, 7, 1, 1, 1, 3, 1, 3, 1, 1, 2, 11, 22, 1, 1, 3, 3, 7, 1, 1, 1, 5, 1, 181, 1, 2, 2, 1, 2, 3, 17, 83, 1, 3, 2, 1, 14, 3, 1, 44, 3, 6, 4, 4, 1, 28, 1, 1, 1, 2, 1, 2, 1, 15, 1, 55, 1, 1, 3, 16, 3, 2, 1, 1, 1, 2, 1, 1, 2, 8, 4, 1, 1, 1, 1, 1, 1, 38

Offset: 0

Clark Kimberling, May 07 2011

Equivalent to the periodic continued fraction [5, r, 5, r, ...] where r=sqrt(5). For geometric interpretations of both continued fractions, see A190287 and A188635.

Cf. A188635, A190287.

ContinuedFraction[(5+Sqrt[25+4Sqrt[5]])/2,120] (* Harvey P. Dale, Jun 22 2022 *)

Offset changed by Andrew Howroyd, Jul 03 2024

cp's OEIS Frontend

A190289 Decimal expansion of (3+sqrt(21))/4.

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A190290 Decimal expansion of (3+sqrt(21))/3.

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A189960 Decimal expansion of (9+27*sqrt(2))/17.

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A189969 Decimal expansion of (7 + sqrt(133))/6, which has periodic continued fractions [3,11,3,1,3,11,3,1,...] and [7/3, 1, 7/3, 1, ...].

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A190181 Decimal expansion of (15+sqrt(465))/12.

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Magma

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A190264 Decimal expansion of (sqrt(89) - 6)/2.

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A190286 Continued fraction of (3+sqrt(9+4r))/2, where r=sqrt(3).

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Extensions

A190288 Continued fraction of (5+sqrt(25+4r))/2, where r=sqrt(5).

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