A188635 -id:A188635 - cp's OEIS frontend

A189965 Continued fraction of (3+x+sqrt(38+6x))/4, where x=sqrt(13).

3, 1, 1, 2, 1, 1, 4, 2, 4, 3, 1, 5, 1, 1, 3, 1, 1, 1, 2, 2, 2, 3, 2, 1, 1, 1, 2, 39, 5, 2, 1, 1, 1, 2, 49, 1, 4, 4, 1, 13, 1, 1, 2, 1, 32, 6, 2, 2, 1, 1, 35, 15, 1, 1, 1, 6, 1, 6, 1, 7, 2, 1, 2, 1, 15, 1, 2, 4, 1, 2, 3, 1, 5, 1, 1, 6, 4, 1, 1, 16, 6, 10, 3, 1, 5, 6, 2, 8, 1, 1, 1, 3, 25, 2, 10, 1, 1, 1, 3, 2, 25, 1, 2, 1, 4, 63, 1, 2, 2, 1, 287, 35, 1, 1, 6, 3, 4, 3, 10, 1

Offset: 0

Clark Kimberling, May 04 2011

See A189964 and A188635.

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

Cf. A189964 (decimal expansion), A188635.

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

Digits:=100: convert(evalf((3+sqrt(13)+sqrt(38+6*sqrt(13)))/4), confrac); # Wesley Ivan Hurt, Dec 12 2013

(See A189964.)
ContinuedFraction[(3 + Sqrt[13] + Sqrt[38 + 6 Sqrt[13]])/4, 100] (* Wesley Ivan Hurt, Dec 12 2013 *)

contfrac((3+sqrt(13)+sqrt(38+sqrt(468)))/4)

Offset changed by Andrew Howroyd, Aug 09 2024

A189966 Decimal expansion of (3+sqrt(33))/4, which has periodic continued fractions [2,5,2,1,2,5,2,1,...] and [3/2, 1, 3/2, 1, ...].

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

Let R denote a rectangle whose shape (i.e., length/width) is (3+sqrt(33))/4. This rectangle can be partitioned into squares in a manner that matches the continued fraction [2,5,2,1,2,5,2,1,2,5,2,1,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [3/2, 1, 3/2, 1, 3/2, ...]. For details, see A188635.

Apart from the first digit, the same as A188939. - R. J. Mathar, May 16 2011

			2.18614066163450716496265286705473232955506611449...

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

Cf. A188635, A188485.

(3+Sqrt(33))/4 // G. C. Greubel, Jan 12 2018

FromContinuedFraction[{3/2, 1, {3/2, 1}}]
ContinuedFraction[%, 25]  (* [2,5,2,1,2,5,2,1,...] *)
RealDigits[N[%%, 120]]  (* A189966 *)
N[%%%, 40]

(3+sqrt(33))/4 \\ G. C. Greubel, Jan 12 2018

A189967 Decimal expansion of (7+sqrt(105))/4, which has periodic continued fractions [4,3,4,1,4,3,4,1...] and [7/2, 1, 7/2, 1, ...].

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

Let R denote a rectangle whose shape (i.e., length/width) is (7+sqrt(105))/4. This rectangle can be partitioned into squares in a manner that matches the continued fraction [4,3,4,1,4,3,4,1...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [7/2, 1, 7/2, 1, ...]. For details, see A188635.

			4.311737691489899595805259670130262997684...

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

Cf. A188635, A189967.

(7+Sqrt(105))/4 // G. C. Greubel, Jan 12 2018

FromContinuedFraction[{7/2, 1, {7/2, 1}}]
ContinuedFraction[%, 25]  (* [4,3,4,1,4,3,4,1...] *)
RealDigits[N[%%, 120]]  (* A189968 *)
N[%%%, 40]

(7+sqrt(105))/4 \\ G. C. Greubel, Jan 12 2018

A189968 Decimal expansion of (5+sqrt(85))/6, which has periodic continued fractions [2,2,1,2,2,1,...] and [5/2, 1, 5/2, 1, ...].

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

Let R denote a rectangle whose shape (i.e., length/width) is (5+sqrt(85))/6. This rectangle can be partitioned into squares in a manner that matches the continued fraction [2,2,1,2,2,1,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [5/2, 1, 5/2, 1, ...]. For details, see A188635.

			2.369924076215481218333712380293798859541...

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

Cf. A188635, A189966.

(5+Sqrt(85))/6 // G. C. Greubel, Jan 12 2018

FromContinuedFraction[{5/2, 1, {5/2, 1}}]
ContinuedFraction[%, 25]  (* [2,2,1,2,2,1,...] *)
RealDigits[N[%%, 120]]  (* A189967 *)
N[%%%, 40]
RealDigits[(5+Sqrt[85])/6,10,120][[1]] (* Harvey P. Dale, Apr 18 2014 *)

(5+sqrt(85))/6 \\ G. C. Greubel, Jan 12 2018

A190182 Decimal expansion of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

The rectangle R whose shape (i.e., length/width) is (1+x+sqrt(8+2x))/4, where x=sqrt(15), can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,1,r,1,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,4,1,3,10,...] at A190183. For details, see A188635.

			2.210275532819020968778971352504887053304...

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

Cf. A188635, A190183, A189970.

[(1 + Sqrt(15) + Sqrt(8 + 2*Sqrt(15)))/4]; // G. C. Greubel, Dec 28 2017

r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, 1, {r, 1, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100] (*A190183*)
RealDigits[N[%%, 120]]    (*A190182*)
N[%%%, 40]
RealDigits[(1 + Sqrt[15] + Sqrt[8 + 2*Sqrt[15]])/4, 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

(1 + sqrt(15) + sqrt(8 + 2*sqrt(15)))/4 \\ G. C. Greubel, Dec 28 2017

A190183 Continued fraction of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

Original entry on oeis.org

2, 4, 1, 3, 10, 1, 3, 1, 1, 2, 66, 1, 4, 2, 1, 1, 48, 5, 1, 1, 2, 1, 1, 1, 8, 2, 1, 1, 4, 16, 2, 2, 1, 4, 1, 3, 1, 3, 1, 11, 1, 1, 8, 16, 1, 1, 1, 10, 1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 1, 30, 1, 1, 2, 1, 1, 8, 13, 1, 1, 6, 6, 1, 6, 1, 1, 2, 2, 10, 1, 2, 7, 9, 2, 4, 7, 3, 1, 2, 2, 1, 2, 5, 4, 2, 3, 2, 3, 2, 1, 3

Offset: 1

Clark Kimberling, May 05 2011

Equivalent to the periodic continued fraction [r,1,1,r,1,1,...] where r=(1+sqrt(5))/2, the golden ratio. For geometric interpretations of both continued fractions, see A190182 and A188635.

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

Cf. A190182, A188635.

ContinuedFraction((1+Sqrt(15)+Sqrt(8+2*Sqrt(15)))/4); // G. C. Greubel, Dec 28 2017

r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, 1, {r, 1, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100] (* A190183 *)
RealDigits[N[%%, 120]]    (* A190182 *)
N[%%%, 40]
ContinuedFraction[(1+Sqrt[15]+Sqrt[8+2Sqrt[15]])/4,100] (* Harvey P. Dale, Apr 29 2013 *)

contfrac((1+sqrt(15)+sqrt(8+2*sqrt(15)))/4) \\ G. C. Greubel, Dec 28 2017

A190256 Decimal expansion of sqrt((3 + x + sqrt(9+6x))/2), where x=sqrt(6).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2011

The rectangle R whose shape (i.e., length/width) is sqrt(3+x+sqrt(9+6x)), where x=sqrt(6), can be partitioned into rectangles of shapes sqrt(3) and sqrt(2) in a manner that matches the periodic continued fraction [sqrt(3), sqrt(2), sqrt(3), sqrt(2), ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,3,1,2,5,2,1,5,95,1,...] at A190257. For details, see A188635.

			2.271281562422994142313058068759726855455...

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

Cf. A190256, A190184, A190185.

[Sqrt((3+Sqrt(6)+Sqrt(9+6*Sqrt(6)))/2)]; // G. C. Greubel, Dec 26 2017

FromContinuedFraction[{3^(1/2), 2^(1/2), {3^(1/2), 2^(1/2)}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190256 *)
RealDigits[N[%%, 120]]     (* A190257 *)
N[%%%, 40]

sqrt((3+sqrt(6)+sqrt(9+6*sqrt(6)))/2) \\ G. C. Greubel, Dec 26 2017

Name corrected by T. D. Noe, Feb 25 2013

A190263 Continued fraction of (3 + sqrt(9 + 12*sqrt(3)))/6.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 2, 1, 1, 1, 1, 8, 2, 17, 2, 3, 10, 2, 23, 1, 4, 1, 2, 1, 4, 1, 2, 35, 4, 1, 1, 1, 2, 5, 4, 1, 1, 3, 17, 3, 2, 1, 3, 1, 3, 1, 1, 10, 3, 1, 13, 1, 1, 1, 4, 1, 2, 2, 2, 1, 2, 15, 3, 2, 5, 6, 2, 1, 15, 132, 4, 2, 1, 1, 19, 1, 4, 1, 2, 5, 2, 16, 2, 1, 15, 5, 2, 10, 13, 1, 1

Offset: 1

Clark Kimberling, May 06 2011

Equivalent to the periodic continued fraction [1, x, 1, x,...], where x=sqrt(3). (See A188635.)

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

Cf. A188635, A190262.

ContinuedFraction((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]
ContinuedFraction[(3 + Sqrt[9 + 12*Sqrt[3]])/6, 100] (* G. C. Greubel, Dec 26 2017 *)

contfrac((3+sqrt(9+sqrt(432)))/6) \\ Charles R Greathouse IV, Jul 29 2011

A190285 Decimal expansion of (3+sqrt(9+4r))/2, where r=sqrt(3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (3+sqrt(9+4r))/2, where r=sqrt(3), can be partitioned into rectangles of shapes 3 and r in a manner that matches the periodic continued fraction [3, r, 3, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [3,2,55,6,1,1,1,9,1,1,1,7,2,...] at A190286. For details, see A188635.

			3.495507656604924503772866679054481005189...

Cf. A188635, A190286.

r=3^(1/2)
FromContinuedFraction[{3, r, {3, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190286 *)
RealDigits[N[%%, 120]]     (* A190285 *)
N[%%%, 40]
RealDigits[(3+Sqrt[9+4Sqrt[3]])/2,10,120][[1]] (* Harvey P. Dale, Oct 19 2021 *)

A190287 Decimal expansion of (5+sqrt(25+4r))/2, where r=sqrt(5).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 07 2011

The rectangle R whose shape (i.e., length/width) is (5+sqrt(25+4r))/2, where r=sqrt(5), can be partitioned into rectangles of shapes 5 and r in a manner that matches the periodic continued fraction [5, r, 5, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [5,2,2,2,1,1,1,10,1,1,2,1,...] at A190288. For details, see A188635.

			5.413085645411028710287065567557494135316...

Cf. A188635, A190288.

r=5^(1/2)
FromContinuedFraction[{5, r, {5, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190288 *)
RealDigits[N[%%, 120]]     (* A190287 *)
N[%%%, 40]

cp's OEIS Frontend

A189965 Continued fraction of (3+x+sqrt(38+6x))/4, where x=sqrt(13).

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A189966 Decimal expansion of (3+sqrt(33))/4, which has periodic continued fractions [2,5,2,1,2,5,2,1,...] and [3/2, 1, 3/2, 1, ...].

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A189967 Decimal expansion of (7+sqrt(105))/4, which has periodic continued fractions [4,3,4,1,4,3,4,1...] and [7/2, 1, 7/2, 1, ...].

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A189968 Decimal expansion of (5+sqrt(85))/6, which has periodic continued fractions [2,2,1,2,2,1,...] and [5/2, 1, 5/2, 1, ...].

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A190182 Decimal expansion of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

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A190183 Continued fraction of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

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Magma

Mathematica

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A190256 Decimal expansion of sqrt((3 + x + sqrt(9+6x))/2), where x=sqrt(6).

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