A344172 -id:A344172 - cp's OEIS frontend

A019952 Decimal expansion of tangent of 54 degrees.

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 36 degrees. - Mohammad K. Azarian, Jun 30 2013
A quartic number with denominator 5. - Charles R Greathouse IV, Aug 27 2017
Conjecture: Product (2/3) * (8/7) * (12/13) * (18/17) * (22/23) * (32/33) * ... * (a_n/b_n) = sqrt(25 + 10*sqrt(5))/5 = tan(3*Pi/10) = A019952, where a_n even, a_n + b_n = a(n), |a_n - b_n| = 1, n >= 0. - Dimitris Valianatos, Feb 14 2020
Also the limiting value of the distance between the lines F(n)*x + F(n+1)*y = 0 and F(n)*x + F(n+1)*y = F(n+2) (where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Apr 03 2021
Decimal expansion of the radius of an inscribed sphere in a rhombic triacontahedron with unit edge length. - Wesley Ivan Hurt, May 11 2021

			1.376381920471173538207209581910887679525899336...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Exact trigonometric constants.
Wikipedia, Rhombic triacontahedron.
Index entries for algebraic numbers, degree 4.

Cf. A019845, A019863, A019934, A090772, A375067.
Cf. A344171 (rhombic triacontahedron surface area).
Cf. A344172 (rhombic triacontahedron volume).
Cf. A344212 (rhombic triacontahedron midradius).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(3*Pi(R)/10); // G. C. Greubel, Nov 22 2018

Digits:=100: evalf(tan(3*Pi/10)); # Wesley Ivan Hurt, Oct 07 2014

RealDigits[Tan[3*Pi/10], 10, 100][[1]] (* Wesley Ivan Hurt, Oct 07 2014 *)
RealDigits[Tan[54 Degree],10,120][[1]] (* Harvey P. Dale, Jul 16 2016 *)

tan(3*Pi/10) \\ Charles R Greathouse IV, Aug 27 2017

from sympy import sqrt
[print(i, end=', ') for i in str(sqrt(1+2/sqrt(5)).n(110)) if i!='.'] # Karl V. Keller, Jr., Jun 19 2020

numerical_approx(tan(3*pi/10), digits=100) # G. C. Greubel, Nov 22 2018

Equals A019863/A019845 = 1/A019934. - R. J. Mathar, Jul 26 2010
The largest positive solution of cos(4*arctan(1/x)) = cos(6*arctan(1/x)). - Thomas Olson, Oct 03 2014
Equals sqrt(25 + 10*sqrt(5))/5. - G. C. Greubel, Nov 22 2018
Equals sqrt(2 + sqrt(5))/5^(1/4). - Burak Muslu, Apr 03 2021
From Wesley Ivan Hurt, May 11 2021: (Start)
Equals phi^2/sqrt(1+phi^2) where phi is the golden ratio.
Equals sqrt(1+2/sqrt(5)). (End)
Equals Product_{k>=1} (1 - (-1)^k/A090772(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A375067. - Hugo Pfoertner, Nov 23 2024

A344212 Decimal expansion of 1 + 1/sqrt(5).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 11 2021

Decimal expansion of the midradius of a rhombic triacontahedron with unit edge length.

Essentially the same sequence of digits as A176453, A134974, A020762 and A010476. - R. J. Mathar, May 16 2021

			1.447213595499957939281834733746255247088123671922305...

Wikipedia, Rhombic triacontahedron.
Index entries for algebraic numbers, degree 2.

Cf. A019952 (rhombic triacontahedron inscribed sphere radius).
Cf. A344171 (rhombic triacontahedron surface area).
Cf. A344172 (rhombic triacontahedron volume).
Cf. A010476, A020762, A081005, A134974, A176453, A187798, A242671.

RealDigits[1 + 1/Sqrt[5], 10, 100][[1]] // Flatten

5^-.5+1 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(5*x^2-10*x+4)[2] \\ Charles R Greathouse IV, Feb 04 2025

From Amiram Eldar, Nov 28 2024: (Start)
Equals 2*A242671 = 1/A187798.
Equals Product_{k>=0} (1 + 1/A081005(k)). (End)

A344171 Decimal expansion of 12*sqrt(5).

Original entry on oeis.org

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

Offset: 2

Wesley Ivan Hurt, May 10 2021

Decimal expansion of the surface area of a rhombic triacontahedron with unit edge length.

			26.83281572999747635691008...

Wikipedia, Rhombic triacontahedron

Cf. A344172 (rhombic triacontahedron volume).
Cf. A344212 (rhombic triacontahedron midradius).
Cf. A019952 (rhombic triacontahedron radius of inscribed sphere).

RealDigits[12 Sqrt[5], 10, 100][[1]] // Flatten

A382008 Decimal expansion of the isoperimetric quotient of a rhombic triacontahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 20 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.88720000254802085800544409398426003785738986572116...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A344171 (surface area), A344172 (volume).

Cf. A000796, A098317, A381684.

First[RealDigits[Pi*(2 + Sqrt[5])/15, 10, 100]]

Equals 36*Pi*A344172^2/(A344171^3).

Equals Pi*(2 + sqrt(5))/15 = A000796*A098317/15.

A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 12 2021

An algebraic number of degree 4. The smaller of the two positive roots of the equation 16*x^4 - 2500*x^2 + 3125 = 0.

			1.12256994144896343110486287949381696894803120580270...

Robert B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, 2012, p. 15.

Herta T. Freitag, Problem 3855, School Science and Mathematics, Vol. 81, No. 4 (1981), p. 352; Solution to Problem 3855 by David A. Blaeuer, ibid., Vol. 82, No. 3 (1982), pp. 265-266.
Mathematics Stack Exchange, Area of a five pointed star, 2014.
Wikipedia, Pentagram.
Index entries for algebraic numbers, degree 4

Cf. A001622, A019845, A019916, A019952, A019970, A094874, A104955, A134974, A179050, A188593, A344172.

RealDigits[5*Sin[Pi/5]/GoldenRatio^2, 10, 100][[1]]

Equals 5*sin(Pi/5)/phi^2, where phi is the golden ratio (A001622).
Equals 5/(cot(Pi/5) + cot(Pi/10)).
Equals 10*tan(Pi/10)/(3 - tan(Pi/10)^2).
Equals (5/2)*sqrt((25 -11*sqrt(5))/2).
Equals 5*(5 - sqrt(5))/(4*sqrt(5 + 2*sqrt(5))) = A094874 * A179050 = 10 * A094874 / A344172.

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A019952 Decimal expansion of tangent of 54 degrees.

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A344212 Decimal expansion of 1 + 1/sqrt(5).

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A344171 Decimal expansion of 12*sqrt(5).

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A382008 Decimal expansion of the isoperimetric quotient of a rhombic triacontahedron.

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A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

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