A300074 -id:A300074 - cp's OEIS frontend

A182007 Decimal expansion of 2*sin(Pi/5).

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

Offset: 1

Stanislav Sykora, Apr 06 2012

The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).
Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.
Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014
This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016
rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.

			1.1755705045849462583374119...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pentagon.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A002193, A003401, A019881, A019845, A063226, A102769, A131595, A188593, A300074.

SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

evalf(2*sin(Pi/5),100); # Muniru A Asiru, Nov 02 2018

RealDigits[2*Sin[Pi/5],10,120][[1]] (* Harvey P. Dale, Sep 29 2012 *)

2*sin(Pi/5) \\ Stanislav Sykora, May 02 2016

Equals sqrt(3-phi).
Equals sqrt((5-sqrt(5))/2). - Jean-François Alcover, May 21 2013
Equals Product_{k>=0} ((10*k + 4)*(10*k + 6))/((10*k + 3)*(10*k + 7)). - Antonio Graciá Llorente, Mar 25 2024
Equals Product_{k>=1} (1 - (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A019845 = 1/A300074. - Hugo Pfoertner, Nov 23 2024

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014
Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022
The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			k2 = 0.723606797749978969640917366873127623544...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Equals (1 + 1/sqrt(5))/2.
Equals 1/A094874. - Michel Marcus, Dec 01 2018
From Amiram Eldar, Feb 11 2022: (Start)
Equals phi/sqrt(5), where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
From Amiram Eldar, Nov 28 2024: (Start)
Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
Equals 1 - A244847. - Amiram Eldar, Mar 18 2025

A375068 Decimal expansion of the sagitta of a regular pentagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 29 2024

			0.1624598481164531630779357061075672324774517357607...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Sagitta
Index entries for algebraic numbers, degree 4.

Cf. A300074 (circumradius), A375067 (apothem), A102771 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375069 (hexagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A019916, A179050, A229760.

First[RealDigits[Tan[Pi/10]/2, 10, 100]]

tan(Pi/10)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(80*x^4-40*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/10)/2 = sqrt(1-2/sqrt(5))/2 = A019916/2.
Equals A300074 - A375067.
Equals A179050/5 = sqrt(A229760)/10. - Hugo Pfoertner, Jul 30 2024

A375067 Decimal expansion of the apothem (inradius) of a regular pentagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 29 2024

			0.688190960235586769103604790955443839762949668...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 4.

Cf. A300074 (circumradius), A375068 (sagitta), A102771 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A019934, A019952, A019863, A131595.

Mathematica
```
First[RealDigits[Cot[Pi/5]/2, 10, 100]]
```

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

Equals cot(Pi/5)/2 = A019952/2.
Equals 1/(2*tan(Pi/5)) = 1/(2*A019934).
Equals sqrt(1/4 + 1/(2*sqrt(5))).
Equals (1/2)*csc(Pi/5)*cos(Pi/5) = A300074*A019863.
Equals A300074 - A375068.
Equals A131595/30. - Hugo Pfoertner, Jul 30 2024

A090773 Numbers that are congruent to {4, 6} mod 10.

Original entry on oeis.org

4, 6, 14, 16, 24, 26, 34, 36, 44, 46, 54, 56, 64, 66, 74, 76, 84, 86, 94, 96, 104, 106, 114, 116, 124, 126, 134, 136, 144, 146, 154, 156, 164, 166, 174, 176, 184, 186, 194, 196, 204, 206, 214, 216, 224, 226, 234, 236, 244, 246, 254, 256, 264, 266, 274, 276, 284

Offset: 1

Giovanni Teofilatto, Feb 07 2004

nonn

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

Cf. A047221, A090298, A179290, A300074.

#+{4,6}&/@(10Range[0,50])//Flatten (* or *) LinearRecurrence[{1,1,-1},{4,6,14},100] (* Harvey P. Dale, Jun 05 2017 *)

a(n) = 2 * A047221(n) = 5*n-5/2-3*(-1)^n/2.
a(n) = 10*n-a(n-1)-10 (with a(1)=4). - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(2+x+2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/10. - Amiram Eldar, Dec 28 2021
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(2*Pi/5) (A179290).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/5)/2 (A300074). (End)

Edited and extended by Ray Chandler, Feb 10 2004

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.

A384036 Decimal expansion of the surface area of a regular pentagonal prism of edge length 1.

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, May 17 2025

			8.4409548011779338455...

Cf. A178809.

Cf. A102771 (volume), A300074 (midradius), A384059 (circumradius).

RealDigits[5 + 1/2 * Sqrt(25 + 10 * Sqrt(5)), 10, 100, 0][[1]]

Equals 5 + (1/2)*sqrt(25 + 10*sqrt(5)).

Minimal polynomial: 16*x^4 - 320*x^3 + 2200*x^2 - 6000*x + 5125. - Stefano Spezia, May 17 2025

A384059 Decimal expansion of the circumradius of a regular pentagonal prism of edge length 1.

Original entry on oeis.org

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

Offset: 0

Kritsada Moomuang, May 18 2025

			0.98671515532598310...

Cf. A102771 (volume), A300074 (midradius), A384036 (surface area).

RealDigits[1/2 * Sqrt(3 + 2/Sqrt(5)), 10, 100, -1][[1]]

Equals (1/2)*sqrt(3 + 2/sqrt(5)).

Minimal polynomial: 80*x^4 - 120*x^2 + 41. - Stefano Spezia, May 18 2025

cp's OEIS Frontend

A182007 Decimal expansion of 2*sin(Pi/5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375068 Decimal expansion of the sagitta of a regular pentagon with unit side length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A375067 Decimal expansion of the apothem (inradius) of a regular pentagon with unit side length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A090773 Numbers that are congruent to {4, 6} mod 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs