A019934 -id:A019934 - 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

A019970 Decimal expansion of tangent of 72 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 18 degrees. - Mohammad K. Azarian, Jun 30 2013
A quartic integer. - Charles R Greathouse IV, Aug 27 2017
Length of the second longest diagonal in a regular 10-gon with unit side. - Mohammed Yaseen, Nov 12 2020

			tan(2*Pi/5) = 3.077683537175253402570290576036909824006702143537792427...

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

Cf. A019881 (sine of 72 degrees).

Cf. A019904, A019934, A019964.

SetDefaultRealField(RealField(100)); Sqrt(5+2*Sqrt(5)); // G. C. Greubel, Nov 21 2018

RealDigits[Tan[72 Degree],10,120][[1]] (* Harvey P. Dale, Apr 30 2012 *)
RealDigits[Sqrt[5 + 2*Sqrt[5]], 10, 100][[1]] (* G. C. Greubel, Nov 21 2018 *)

tan(2*Pi/5) \\ Charles R Greathouse IV, Aug 27 2017

numerical_approx(tan(2*pi/5), digits=100) # G. C. Greubel, Nov 21 2018

Equals sqrt(5 + 2*sqrt(5)). - R. J. Mathar, Jun 18 2006
Equals tan(66 degrees) + tan(36 degrees) + tan(6 degrees). - Amiram Eldar, Apr 07 2022
Largest positive of the 4 real-valued roots of x^4-10*x^2+5=0. - R. J. Mathar, Aug 31 2025

A063226 Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).

Original entry on oeis.org

3, 7, 13, 17, 23, 27, 33, 37, 43, 47, 53, 57, 63, 67, 73, 77, 83, 87, 93, 97, 103, 107, 113, 117, 123, 127, 133, 137, 143, 147, 153, 157, 163, 167, 173, 177, 183, 187, 193, 197, 203, 207, 213, 217, 223, 227, 233, 237, 243, 247

Offset: 1

N. J. A. Sloane, Jul 10 2001

Also, dimension of the space of weight 2n cuspidal newforms for Gamma_0(88). - N. J. A. Sloane, Nov 24 2016
First differences are 4,6,4,6,4,6.... Also  values of k such that k^(10*n) mod 10 = 8*(n mod 2)+1. - Gary Detlefs, Jul 04 2014
In other words, numbers n such that n^(2+4*k) + 1 is divisible by 10, for k >= 0. - Altug Alkan, Mar 30 2016
The rational generating function, the periodic first differences and Greubel's closed form are an immediate consequence of the structure of formula given by [Martin]. - R. J. Mathar, Apr 09 2016
A quasipolynomial of order 2 and degree 1: a(n) = 5n - 3 if n is even and 5n - 2 if n is odd. - Charles R Greathouse IV, Nov 03 2021
Numbers that are congruent to {3, 7} mod 10. - Amiram Eldar, Nov 23 2024

G. C. Greubel, Table of n, a(n) for n = 1..10000
Greg Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331, Theorem 1.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A017305 (bisection), A017353 (bisection), A019934, A182007.

Maple
```
# see A063195
```

Table[4 Floor[n/2] + 6 Floor[(n - 1)/2] + 3, {n, 50}] (* or *)
Table[SeriesCoefficient[3 x - x^2 (-7 - 6 x + 3 x^2)/((1 + x) (x - 1)^2), {x, 0, n}], {n, 50}] (* Michael De Vlieger, Mar 30 2016 *)
LinearRecurrence[{1, 1, -1}, {3, 7, 13}, 100] (* G. C. Greubel, Mar 30 2016 *)

my(x='x+O('x^99)); Vec(3*x-x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2)) \\ Altug Alkan, Mar 31 2016

a(n)=5*n-3+n%2 \\ Charles R Greathouse IV, Mar 31 2016

a(n) = 4*floor(n/2) + 6*floor((n-1)/2) + 3. - Gary Detlefs, Jul 04 2014
G.f.: 3*x - x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2). - R. J. Mathar, Jul 15 2015
From G. C. Greubel, Mar 30 2016: (Start)
a(n) = (1/2)*(10*n - 5 - (-1)^n).
E.g.f.: (5*x + 3)*cosh(x) + (5*x + 2)*sinh(x). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5-2*sqrt(5))*Pi/10. - Amiram Eldar, Sep 26 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(Pi/5) (A182007).
Product_{n>=1} (1 + (-1)^n/a(n)) = tan(Pi/5) (A019934). (End)

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

A165225 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 5a(n-2) for n > 1.

Original entry on oeis.org

1, 5, 45, 425, 4025, 38125, 361125, 3420625, 32400625, 306903125, 2907028125, 27535765625, 260822515625, 2470546328125, 23401350703125, 221660775390625, 2099601000390625, 19887706126953125, 188379056267578125

Offset: 0

Philippe Deléham, Sep 09 2009

Sum_{k=1..(m-1)/2} tan^(2n) (k*Pi/m) is an integer when m >= 3 is an odd integer (see AMM and Crux Mathematicorum links); twice this sequence is the particular case m = 5. - Bernard Schott, Apr 25 2022

Harvey P. Dale, Table of n, a(n) for n = 0..1000.
Michel Bataille and Li Zhou, A Combinatorial Sum Goes on Tangent, The American Mathematical Monthly, Vol. 112, No. 7 (Aug. - Sep., 2005), Problem 11044, pp. 657-659.
D. J. Smeenk, Problem 3452, Crux Mathematicorum, Vol. 35, No. 5 (2009), pp. 325 and 327; Solution to Problem 3452 by Roy Barbara, ibid., Vol. 36, No. 5 (2010), pp. 341-342.
Index entries for linear recurrences with constant coefficients, signature (10,-5).

Cf. A019934, A019970.

Similar with: A000244 (m=3), 2*this sequence (m=5), A108716 (m=7), A353410 (m=9), A275546 (m=11), A353411 (m=13).

LinearRecurrence[{10,-5},{1,5},30] (* Harvey P. Dale, Dec 23 2019 *)

Limit_{n->oo} a(n+1)/a(n) = 5 + 2*sqrt(5) = 9.47213595...
G.f.: (1-5x)/(1-10x+5x^2).
a(n) = ((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/2. - Klaus Brockhaus, Sep 25 2009
a(n) = (tan(Pi/5)^(2*n) + tan(2*Pi/5)^(2*n))/2 (Smeenk, 2009). - Amiram Eldar, Apr 03 2022

More terms from Klaus Brockhaus, Sep 25 2009

A190987 a(n) = 10a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 95, 900, 8525, 80750, 764875, 7245000, 68625625, 650031250, 6157184375, 58321687500, 552430953125, 5232701093750, 49564856171875, 469485056250000, 4447026281640625, 42122837535156250, 398993243943359375, 3779318251757812500, 35798216297861328125

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-5).

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A019934 (sqrt(5-2*sqrt(5))), A019952 (sqrt(5+2*sqrt(5))).

[Round(5^((n-1)/2)*Evaluate(ChebyshevU(n), Sqrt(5))): n in [0..30]]; // G. C. Greubel, Sep 07 2022

Mathematica
```
LinearRecurrence[{10,-5}, {0,1}, 50]
```

A190987 = BinaryRecurrenceSequence(10, -5, 0, 1)
[A190987(n) for n in (0..30)] # G. C. Greubel, Sep 07 2022

G.f.: x/(1 - 10*x + 5*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (1/(2*sqrt(5)))*exp(5*x)*sinh(2*sqrt(5)*x). - G. C. Greubel, Sep 07 2022

A343057 Decimal expansion of tan(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.098491403357164253077197...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8.

Cf. A343055 (sin(Pi/32)), A343056 (cos(Pi/32)).

tan(Pi/m): A002194 (m=3), A019934 (m=5), A020760 (m=6), A343058 (m=7), A188582 (m=8), A019918 (m=9), A019916 (m=10), A019913 (m=12), A343059 (m=14), A019910 (m=15), A343060 (m=16), A343061 (m=17), A019908 (m=18), A019907 (m=20), A343062 (m=24), A019904 (m=30), A343057 (m=32), A019903 (m=36).

R:= RealField(125); Tan(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Tan[Pi/32], 10, 120, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/32)
```

sqrt((2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))))

numerical_approx(tan(pi/32), digits=125) # G. C. Greubel, Sep 30 2022

Equals sqrt( (2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))) ).

A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Oct 10 2022

The perimeter of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			5.5055276818846941...

Cf. A019934, A019952, A019970, A121570, A179290, A204188, A356869.

Maple
```
sqrt(16 + 32 / sqrt(5));
```
Mathematica
```
Sqrt[16 + 32/Sqrt[5]]
```
PARI
```
sqrt(16 + 32 / sqrt(5))
```

Equals (4 / sqrt(5)) * sqrt(5 + 2 * sqrt(5)) = A356869 * A019970.
Equals sqrt(5 + 2 * sqrt(5)) / (sqrt(5) / 4) = A019970 / A204188.
Equals 4 * sqrt(1 + 2 / sqrt(5)) = 4 * A019952.
Equals 4 / sqrt(5 - 2 * sqrt(5)) = 4 / A019934.

cp's OEIS Frontend

A019952 Decimal expansion of tangent of 54 degrees.

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A019970 Decimal expansion of tangent of 72 degrees.

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A063226 Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).

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

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A165225 a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 5*a(n-2) for n > 1.

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A190987 a(n) = 10*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

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A343057 Decimal expansion of tan(Pi/32).

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A165225 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 5a(n-2) for n > 1.

A190987 a(n) = 10a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.