A020760 -id:A020760 - cp's OEIS frontend

A165952 Decimal expansion of 2sqrt(3)/(3Pi).

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a cube to the volume of the circumscribed sphere (which has circumradius a*sqrt(3)/2 = a*A010527, where a is the cube's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165953, and A165954. A063723 shows the order of these by size.

			0.3675525969478613663408843322086462942649243202444271018662440135273535356...

Eric Weisstein's World of Mathematics, Cube.
Index entries for transcendental numbers

Cf. A000796, A002194, A165922, A049541, A165953, A165954, A063723, A020760, A020832.

RealDigits[(2*Sqrt[3])/(3Pi),10,120][[1]] (* Harvey P. Dale, Oct 08 2012 *)

PARI
```
2*sqrt(3)/(3*Pi)
```

2*sqrt(3)/(3*Pi) = 2*A002194/(3*A000796) = 3*A165922 = (2*sqrt(3)/3)*A049541 = 10*A020832*A049541 = 2*A020760*A049541.

A132367 Period 6: repeat [1, 1, 2, -1, -1, -2].

Original entry on oeis.org

1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2, 1, 1, 2, -1, -1, -2

Offset: 0

Paul Curtz, Nov 09 2007

Nonsimple continued fraction expansion of 1+1/sqrt(3) = 1 + A020760. - R. J. Mathar, Mar 08 2012

Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A020760, A061347, A100051, A100063, A122876.

&cat[[1, 1, 2, -1, -1, -2]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A132367:=n->[1, 1, 2, -1, -1, -2][(n mod 6)+1]: seq(A132367(n), n=0..100); # Wesley Ivan Hurt, Jun 19 2016

PadRight[{}, 120, {1,1,2,-1,-1,-2}] (* Harvey P. Dale, Jul 28 2012 *)

a(n)=[1,1,2,-1,-1,-2][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n) = cos(Pi*n/3)/3+sqrt(3)*sin(Pi*n/3)+2*(-1)^n/3. - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 19 2016: (Start)
G.f.: (1+x+2*x^2)/(1+x^3).
a(n) + a(n-3) = 0 for n>2. (End)

A374957 Decimal expansion of the circumradius of a regular heptagon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 26 2024

			1.15238243548124325262057511177342755672225094380...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 6.

Cf. A374971 (apothem), A374972 (sagitta), A178817 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A285871 (octagon), A375151 (9-gon), A001622 (10-gon), A375190 (11-gon), A188887 (12-gon).
Cf. A073052, A121598, A272487.

Mathematica
```
First[RealDigits[Csc[Pi/7]/2, 10, 100]]
```

 5/sin(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/7)/2 = A121598/2.
Equals 1/(2*sin(Pi/7)) = 1/A272487.
Equals A374971/cos(Pi/7) = A374971/A073052.
Largest of the 6 real-valued roots of 7*x^6-14*x^4+7*x^2-1=0. - R. J. Mathar, Aug 29 2025

A375151 Decimal expansion of the circumradius of a regular 9-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2024

			1.46190220008154362611637720668314585193675283...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 6.

Cf. A375152 (apothem), A375153 (sagitta), A256853 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A374957 (heptagon), A285871 (octagon), A001622 (10-gon), A375190 (11-gon), A188887 (12-gon)
Cf. A019879, A121602, A272488.

Mathematica
```
First[RealDigits[Csc[Pi/9]/2, 10, 100]]
```

 5/sin(Pi/9) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/9)/2 = A121602/2.
Equals 1/(2*sin(Pi/9)) = 1/A272488.
Equals A375152/cos(Pi/9) = A375152/A019879.
Equals A375152 + A375153.
Largest of the 6 real-valued roots of 3*x^6-9*x^4+6*x^2-1=0. - R. J. Mathar, Aug 29 2025

A375190 Decimal expansion of the circumradius of a regular 11-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 04 2024

			1.774732766442111662856831961168975846105376382123...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 10.

Cf. A375191 (apothem), A375192 (sagitta), A256854 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A374957 (heptagon), A285871 (octagon), A375151 (9-gon), A001622 (10-gon), A188887 (12-gon).
Cf. A272489.

First[RealDigits[Csc[Pi/11]/2, 10, 100]]

 5/sin(Pi/11) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/11)/2.
Equals 1/(2*sin(Pi/11)) = 1/A272489.
Equals A375191/cos(Pi/11).
Equals A375191 + A375192.

A023116 Signature sequence of 1/sqrt(3) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

1/sqrt(3) =  0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495251...
which is different from the constant x that defines A084822, which is
  x=0.5772846089557105648944851585150204938530753765812706743158491735333298369
so the two sequences are different.  Where do they first differ? - N. J. A. Sloane, Jan 20 2023
The first difference is A084822(2793) = 1 and a(2793) = 57. - Paul D. Hanna and Michael S. Branicky, Jan 21 2023

Clark Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
Clark Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A084822, A020760.

A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 08 2009

			1.11535507165041054076705837455583093794582718446458...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, 1996, 4.1.49.

Cf. A010503, A020760, A020763, A092259.

Maple
```
1/2*(1+1/3*3^(1/2))*2^(1/2);
```

RealDigits[1/Sqrt[2] + 1/Sqrt[6], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

1/sqrt(6) + 1/sqrt(2) \\ Michel Marcus, Jan 15 2021

Equals (1+A020760)*A010503.
Equals A020763 + A010503. - Artur Jasinski, Dec 20 2020
The minimal polynomial is 9*x^4 - 12*x^2 + 1. - Joerg Arndt, Sep 20 2023
Equals 2F1(1/4,3/4; 1/2; 1/4). - R. J. Mathar, Aug 02 2024
Equals Product_{k>=1} (1 - (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

Typo in definition corrected by R. J. Mathar, Feb 09 2009

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.

			0.6649088942053266431144284467086337161648765805556919381057592605722964718...

Eric Weisstein's World of Mathematics, Dodecahedron.
Index entries for transcendental numbers.

Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.

RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi),10,120][[1]] (* Harvey P. Dale, Feb 16 2018 *)

PARI
```
(5*sqrt(3)+sqrt(15))/(6*Pi)
```

Equals (5*A002194 + A010472)/(6*A000796).
Equals (5*A002194 + A010472)*A049541/6.
Equals (10*A010527 + A010472)*A049541/6.
Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).
Equals (5 + A002163)*A049541*A020760/2.

A020772 Decimal expansion of 1/sqrt(15).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

1/sqrt(15) = 0.258198889747161125678617693318826640722194780352772721772504917740898872796... [Vladimir Joseph Stephan Orlovsky, May 30 2010]

Ivan Panchenko, Table of n, a(n) for n = 0..1000

RealDigits[N[1/Sqrt[15],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals 1/A010472 = A020760 * A020762. - R. J. Mathar, Nov 19 2024

A084822 Signature sequence of x, where x=0.577284608955710564894... (A084823) is the unique number between 0 and 1 having the property that the signature sequence of x is equal to the continued fraction expansion of x.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 5, 1, 4, 3, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 9, 5, 1, 8, 4, 7, 3, 6, 2, 9, 5, 1, 8, 4, 7, 3, 10, 6, 2, 9, 5, 1, 8, 4, 7, 3, 10, 6, 2, 9, 5, 1, 8, 4, 11, 7, 3

Offset: 1

Paul D. Hanna, Jun 04 2003

The initial terms are close to those of A023116 since x is close to 1/sqrt(3)=A020760. (See A023116.) - R. J. Mathar, Sep 17 2008
Where do they first differ? - N. J. A. Sloane, Jan 20 2023
The first difference is a(2793) = 1 and A023116(2793) = 57. - Paul D. Hanna and Michael S. Branicky, Jan 21 2023

			Given x = 0.5772846089557105648944851585150204938530... (A084823), the continued fraction of x equals the signature sequence of x.
To obtain the signature sequence of x, arrange the numbers i+j*x (i,j >= 1) in increasing order like so:
[1+1*x, 1+2*x, 2+1*x, 1+3*x, 2+2*x, 1+4*x, 3+1*x, 2+3*x, 1+5*x, 3+2*x, 2+4*x, 1+6*x, 4+1*x, 3+3*x, 2+5*x, 1+7*x, 4+2*x, 3+4*x, 2+6*x, 5+1*x, 1+8*x, 4+3*x, 3+5*x, 2+7*x, 5+2*x, 1+9*x, 4+4*x, 3+6*x, 6+1*x, 2+8*x, 5+3*x, 1+10*x, 4+5*x, 3+7*x, 6+2*x, 2+9*x, 5+4*x, 1+11*x, 4+6*x, 7+1*x, 3+8*x, 6+3*x, 2+10*x, 5+5*x, 1+12*x, 4+7*x, 7+2*x, 3+9*x, 6+4*x, 2+11*x, ...];
then the sequence of i's is the signature of x, and forms this sequence.

Paul D. Hanna, Table of n, a(n) for n = 1..2000
Index entries for sequences related to signature sequences

Cf. A084823 (decimal expansion).

cp's OEIS Frontend

A165952 Decimal expansion of 2*sqrt(3)/(3*Pi).

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A132367 Period 6: repeat [1, 1, 2, -1, -1, -2].

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A374957 Decimal expansion of the circumradius of a regular heptagon with unit side length.

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A375151 Decimal expansion of the circumradius of a regular 9-gon with unit side length.

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A375190 Decimal expansion of the circumradius of a regular 11-gon with unit side length.

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A023116 Signature sequence of 1/sqrt(3) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

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A145439 Decimal expansion of Sum_{k>=0} binomial(4*k, 2*k)/2^(6*k).

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Extensions

A165952 Decimal expansion of 2sqrt(3)/(3Pi).

A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).