A182007 -id:A182007 - cp's OEIS frontend

A005531 Decimal expansion of fifth root of 2.

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

Offset: 1

N. J. A. Sloane

The sine of 2017 times this number is the near-integer 0.999999999999999978567771261.... - Alonso del Arte, May 03 2013

With the present number r = 2^(1/5) and the golden section phi = A001622 the other (complex) roots of x^5 - 2 are given by x1 = r*exp(2*Pi*i/5) = r*(phi - 1 + sqrt(2 + phi)*i)/2 = r*(A001622 - 1 + A188593*i)/2 = 0.3549673131... + 1.0924770557...*i, x2 = r*exp(4*Pi*i/5) = r*(-phi + sqrt(3 - phi)*i)/2 = r*(-A001622 + A182007*i)/2 = -0.9293164906... + 0.6751879523...*i, and their complex conjugates. - Wolfdieter Lang, Dec 06 2022

			1.148698354997035006798626946777927589443850889097797505513711118493603....

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 5.

Cf. A002950 (continued fraction).
Cf. A002580 (cube root of 2).
Cf. A001622, A182007, A188593.

RealDigits[N[2^(1/5),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)
RealDigits[Surd[2,5],10,120][[1]] (* Harvey P. Dale, May 08 2025 *)

{ default(realprecision, 20080); x=2^(1/5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b005531.txt", n, " ", d)); } \\ Harry J. Smith, May 12 2009

Equals Product_{k>=0} (1 + (-1)^k/(5*k + 4)). - Amiram Eldar, Jul 25 2020
From Peter Bala, Mar 02 2022: (Start)
Equals (3/2)*Sum_{n >= 0} (1/(5*n+2) - 1/(5*n-3))*binomial(1/5,n). Cf. A002580.
Equals (5/4)*hypergeom([-1/5, -3/5], [7/5], -1). (End)

More terms from Olaf Voß, Feb 13 2008

A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Feb 25 2014

Equals phi^2/(2*xi), where phi is the golden ratio (A001622, 2*cos(Pi/5)) and xi is its associate (A182007, 2*sin(Pi/5)).

			1.1135163644116067351943750394869493758831503698864877726012080...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A182007, A019863, A019863, A019952, A374771 (sphere volume).

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A179294 (icosahedron).

RealDigits[ Cos[Pi/5]^2 / Sin[Pi/5], 10, 111][[1]] (* Or *)
RealDigits[ Sqrt[5/8 + 11/(8 Sqrt[5])], 10, 111][[1]] (* Robert G. Wilson v, Feb 28 2014 *)

PARI
```
sqrt(250+110*sqrt(5))/20
```

Equals A001622^2/A182007 = (cos(Pi/5))^2/sin(Pi/5) = A019863^2/A019845 = cos(Pi/5)*cotan(Pi/5) = A019863*A019952 = 1/sin(Pi/5) - sin(Pi/5) = A019845^(-1) - A019845 = sqrt(250+110*sqrt(5))/20.

A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.
This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)

			0.415823381635518674203484568810250332433169521255447672814363947...

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for sequences related to Chebyshev polynomials.
Index entries for algebraic numbers, degree 8.

Cf. A003401, A019434, A127672, A302711, A302716.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/15], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/15)
```

Equals 2*sin(Pi/m) for m=15, 2*A019821.
Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.
Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - Wolfdieter Lang, Apr 29 2018

A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

Like all m-gons with m equal to a power of 2 (see A003401 and A000079), this is a constructible number.

			0.390180644032256535696569736954044481855383235503909615509004...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A000079, A003401.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/16], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/16)
```

Equals 2*sin(Pi/m) for m=16, 2*A232738. Equals also sqrt(2-sqrt(2+sqrt(2))).

A272536 Decimal expansion of the edge length of a regular 20-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

Since 20-gon is constructible (see A003401), this is a constructible number.

			0.3128689300804617380202106389343337846277997841713215801692826921...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A003401.
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17).
Cf. A019818.

RealDigits[N[2Sin[Pi/20], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/20)
```

Equals 2*sin(Pi/20) = 2*A019818.
Equals also (sqrt(2)+sqrt(10)-2*sqrt(5-sqrt(5)))/4.
Equals i^(9/10) + i^(-9/10). - Gary W. Adamson, Jul 08 2022

A385445 Decimal expansion of (-1 + 3phi)sqrt(3 - phi), with the golden section phi = A001622.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jul 01 2025

This constant c gives the real part of -2*11*Z = (c + d*i), where Z is the (finite) fixed point of a complex function w (of the loxodromic type) mapping iteratively the vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumscribed by the unit circle with center at the origin, and D_1 = i, D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-(3*Pi/5)*i) = -((2 - phi) + s*i)/2 and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D_2, D_3) of the first triangle, also called s_1).
The imaginary part of -2*11*Z is d = -7 + 10*phi = A385446.
If the fixed point Z = -(0.20594... + 0.41728...*i) is chosen as origin then the loxodromic map is W(z') = a*z' (where z' = z - Z and W(z') = w(z'+Z) - Z).
For details see the linked paper, eqs.(5a,b) for w(z), eq.(6) for Z and eq.(7) for W(z'). (In eq.(5b) the i is missing in the exponent.) The nesting of golden triangles as shown in Fig. 1 of the link leads to the fixed point Z.
The vertices of the nested golden triangles can be connected by a spiral built of circular sections with angle 108 degrees, centered at vertices D_{n+3} and shrinking radii s_n =(- 1 + phi)^(n-1)*s. Note that the curvature of this spiral is not continuous.
The length(Z, D_n) =: rho_n of the spokes of the spiral is (-1 + phi)^(n-1)*rho_1, with sqrt(11)*rho_1 = sqrt(8 + 9*phi) = sqrt(5 + 7*phi)*s = A385447.
For the length ratio rho_1/s see A385448.
The logarithmic spiral connecting the vertices D_{n+1} is given in polar coordinates by rho(Phi) = rho_1 * exp((-(5/(3*Pi)) * log(phi)*Phi), with the vertices obtained in polar coordinates for Phi = Phi_n = (3*Pi/5)*n, namely rho(Phi_n) = rho_{n+1}, for n >= 0. For log(phi) see A002390. Note that the nonnegative x-axis is now along Z, D_1. The angle(Z, D1, D4) =: gamma is given by arctan((18 - 11*phi)/s) = arcsin(rho_4 / 2) = 0.169860704... or 9.7323... degrees. See Fig. 3 of the linked paper.

			4.5307685931859751743612240909981473232388869294682093...

Wolfdieter Lang, On a Conformal Mapping of Golden Triangles, Papua New Guinea Journal of Maths. 1991, Vol.2, No:2, pp. 12 - 18 (with corrections).

Cf. A001622, A002390, A090550, A182007, A385446, A385447, A385448.

RealDigits[Sqrt[27 - 4*GoldenRatio], 10, 120][[1]] (* Amiram Eldar, Jul 02 2025 *)

Equals (-1 + 3*phi)*sqrt(3 - phi) = (A090550 - 2)*A182007.

Equals sqrt(27 - 4*phi).

A258403 Decimal expansion of the area of the regular 10-gon (decagon) of circumradius = 1.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 29 2015

Quartic number of degree 4 and denominator 2; minimal polynomial 16x^4 - 500x^2 + 3125. - Charles R Greathouse IV, Apr 20 2016

			2.9389262614623656458435297731953638429882621882157299553613624...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Eric Weisstein's MathWorld, Decagon
Wikipedia, Decagon

Cf. A104954 (triangle), A104955 (pentagon), A104956 (hexagon), A104957 (heptagon).

Cf. A178816 (area of decagon with edge length 1). A182007.

RealDigits[(5/2)*Sqrt[(5 - Sqrt[5])/2], 10, 101] // First

(5/2)*sqrt((5 - sqrt(5))/2) \\ Michel Marcus, May 29 2015

Equals (5/2)*sqrt((5-sqrt(5))/2).
Area formulas from triangle to dodecagon, with circumradius 1:
   n-gon    area(n) = (1/2)*n*sin(2*Pi/n)
   3-gon    (3*sqrt(3))/4
   4-gon    2
   5-gon    (5/4)*sqrt((5+sqrt(5))/2)
   6-gon    (3*sqrt(3))/2
   7-gon    (7/2)*cos((3*Pi)/14)
   8-gon    2*sqrt(2)
   9-gon    (9/2)*sin((2*Pi)/9)
  10-gon    (5/2)*sqrt((5-sqrt(5))/2)
  11-gon    (11/2)*sin((2*Pi)/11)
  12-gon    3
This constant is (5/2)*A182007. - Wolfdieter Lang, May 08 2018

A335137 a(n) = floor(nIm(2e^(i*Pi/5))).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72

Offset: 1

Karl V. Keller, Jr., May 24 2020

nonn

This is the Beatty sequence for imaginary part of 2*e^(i*Pi/5).
Im(2*e^(i*Pi/5)) = A182007 = 1.1755705045849462583374119... = 2*sin(Pi/5).
The real part of floor(n*2*e^(i*Pi/5)) is A000201 (floor(n*phi)).
Re(2*e^(i*Pi/5)) = A001622 = phi = (1 + sqrt(5))/2.
For n < 57, a(n) = A109234(n).

			For n = 3, floor(3*1.17557) = 3.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A000201, A001622, A109234, A182007.

Array[Floor[# Im[2 E^(I*Pi/5)]] &, 62] (* Michael De Vlieger, May 24 2020 *)

from sympy import floor, im, exp, I, pi
for n in range(1, 101): print(floor(n*im(2*exp(I*pi/5))), end=', ')

A385446 Decimal expansion of -7 + 10*phi, with the golden section phi = A001622.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jul 01 2025

This constant d gives the imaginary part of -2*11*Z = c + d*i, where Z is the fixed point of a complex function w (of the loxodromic type) mapping vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumcribed by the unit circle with center at the origin, and D_1 = i (the complex unit), D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-3*Pi*i/5) = -((2 - phi) + s*i)/2  and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D2, D3) of the first triangle).
The real part c = (-1 + 3*phi)*s is given in A385445.
For details see A385445, and eqs.(5a,b) of the linked paper there.

			9.18033988749894848204586834365638117720309179805762862...

Cf. A001622, A182007, A385445.

RealDigits[10*GoldenRatio - 7, 10, 120][[1]] (* Amiram Eldar, Jul 02 2025 *)

Equals -7 + 10*phi, an integer in the quadratic number field Q(sqrt(5)).

Equals 10*A176055-12 = 10*A104457-17 = 10*A001622-7 . - R. J. Mathar, Jul 06 2025

A385447 Decimal expansion of sqrt(8 + 9*phi), with the golden section A001622.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jul 01 2025

This constant c gives sqrt(11)*rho_1, where rho_1 = length(Z, D_1), with the fixed point Z = -(A385445 + A385446*i)/(2*11) of the complex map w given in A385445 and D_1 = i.

See A385445 for details and the linked paper, eq. (7b).

			4.7499795682454312672720369270371548927...

Cf. A001622, A182007, A385445, A385446.

RealDigits[Sqrt[8 + 9*GoldenRatio], 10, 120][[1]] (* Amiram Eldar, Jul 02 2025 *)

Equals sqrt(8 + 9*phi) = sqrt(5 + 7*phi)*s, with s = A182007, the length of the side of a regular pentagon circumcribed by a unit circle.

cp's OEIS Frontend

A005531 Decimal expansion of fifth root of 2.

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A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

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A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

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A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

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A272536 Decimal expansion of the edge length of a regular 20-gon with unit circumradius.

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A385445 Decimal expansion of (-1 + 3*phi)*sqrt(3 - phi), with the golden section phi = A001622.

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A258403 Decimal expansion of the area of the regular 10-gon (decagon) of circumradius = 1.

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A385445 Decimal expansion of (-1 + 3phi)sqrt(3 - phi), with the golden section phi = A001622.

A335137 a(n) = floor(nIm(2e^(i*Pi/5))).