A090772 -id:A090772 - 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

A090298 Permutation of natural numbers generated by 5-row array shown below.

Original entry on oeis.org

1, 9, 2, 11, 8, 3, 19, 12, 7, 4, 21, 18, 13, 6, 5, 29, 22, 17, 14, 10, 31, 28, 23, 16, 15, 39, 32, 27, 24, 20, 41, 38, 33, 26, 25, 49, 42, 37, 34, 30, 51, 48, 43, 36, 35, 59, 52, 47, 44, 40, 61, 58, 53, 46, 45, 69, 62, 57, 54, 50, 71, 68, 63, 56, 55, 79, 72, 67, 64, 60, 81, 78

Offset: 1

Giovanni Teofilatto, Jan 25 2004

9 11 19 21 29 31 39... (A090771)
8 12 18 22 28 32 38... (A090772)
7 13 17 23 27 33 37... (A063226)
6 14 16 24 26 34 36... (A090773)
10 15 20 25 30 35 40... (A008587, excluding initial term)
-----------------------------------------------------------
For such arrays A_k, here A_5, see a W. Lang comment on A113807, the A_7 case. However, in order to obtain A_5 one should take the last row as the first one after adding a 0 in front (thus getting a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Cf. A088520, A090243, A092260, A113807.

More terms from Ray Chandler, Feb 01 2004

A247643 a(n) = ( 10n(n+1)+(2n+1)(-1)^n+7 )/8.

Original entry on oeis.org

1, 3, 9, 15, 27, 37, 55, 69, 93, 111, 141, 163, 199, 225, 267, 297, 345, 379, 433, 471, 531, 573, 639, 685, 757, 807, 885, 939, 1023, 1081, 1171, 1233, 1329, 1395, 1497, 1567, 1675, 1749, 1863, 1941, 2061, 2143, 2269, 2355, 2487, 2577, 2715, 2809, 2953, 3051

Offset: 0

N. J. A. Sloane, Sep 23 2014

From Paul Curtz, Jan 01 2020: (Start)
In the following pentagonal spiral of odd numbers
                     101
                  99  61  63
              97  59  31  33  65
          95  57  29  11  13  35  67
      93  55  27   9   1   3  15  37  69
        91  53  25   7   5  17  39  71
          89  51  23  21  19  41  73
            87  49  47  45  43  75
              85  83  81  79  77
the terms of this sequence appear on the x axis. A062786 and A172043 are in the spiral as well. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

A diagonal of triangle in A247646.

Cf. A005408, A062786, A085787, A090772, A172043.

Maple
```
f:=n->(10*n*(n+1)+(2*n+1)*(-1)^n+7)/8;
```

Table[(10 n (n + 1) + (2 n + 1) (-1)^n + 7)/8, {n, 0, 60}] (* Vincenzo Librandi, Sep 26 2014 *)

Vec(-(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 25 2014

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Colin Barker, Sep 25 2014
G.f.: -(x^4+2*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Sep 25 2014
From Paul Curtz, Jan 01 2020: (Start)
a(n) = 1 + 2*A085787(n).
a(n+1) = a(n-1) + A090772(n+1). (End)
E.g.f.: (1/4)*((1 + x)*(4 + 5*x)*cosh(x) + (3 + x*(11 + 5*x))*sinh(x)). - Stefano Spezia, Jan 01 2020

More terms from Colin Barker, Sep 25 2014

A063225 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 62 ).

Original entry on oeis.org

3, 8, 12, 18, 22, 28, 32, 38, 42, 48, 52, 58, 62, 68, 72, 78, 82, 88, 92, 98, 102, 108, 112, 118, 122, 128, 132, 138, 142, 148, 152, 158, 162, 168, 172, 178, 182, 188, 192, 198, 202, 208, 212, 218, 222, 228, 232, 238, 242, 248

Offset: 1

N. J. A. Sloane, Jul 10 2001

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).

Essentially the same as A090772.

G.f.: 3x -2*x^2*(-4-2*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 15 2015

A317657 Numbers congruent to {15, 75, 95} mod 100.

Original entry on oeis.org

15, 75, 95, 115, 175, 195, 215, 275, 295, 315, 375, 395, 415, 475, 495, 515, 575, 595, 615, 675, 695, 715, 775, 795, 815, 875, 895, 915, 975, 995, 1015, 1075, 1095, 1115, 1175, 1195, 1215, 1275, 1295, 1315, 1375, 1395, 1415, 1475, 1495, 1515

Offset: 1

Paul Curtz, Aug 03 2018

Numbers written in French ending in "quinze".

a(n) = 5 * (3, 15, 19, 23, 35, 39, 43, 55, 59, ... ).

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

Cf. A008587, A008854, A047205, A090772, A290781, A317633.

Filtered([0..1520], n->n mod 100=15 or n mod 100=75 or n mod 100=95); # Muniru A Asiru, Aug 29 2018

select(n->modp(n,100)=15 or modp(n,100)=75 or modp(n,100)=95,[$0..1520]); # Muniru A Asiru, Aug 29 2018

Rest@ CoefficientList[Series[(5 x (x^3 + 4 x^2 + 12 x + 3))/((x^2 + x + 1) (x - 1)^2), {x, 0, 46}], x] (* Michael De Vlieger, Aug 05 2018 *)
Table[100*n/3 - 80*Sin[2*n*Pi/3]/(3*Sqrt[3]) - 5,{n,1,46}] (* Stefano Spezia, Aug 29 2018 *)

a(n) = 10*A317633(n) + 5.
a(n) = a(n-3) + 100, a(1) = 15, a(2) = 75, a(3) = 95.
From Franck Maminirina Ramaharo, Aug 05 2018: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4), n>4.
a(n) = A290781(A047205(n)).
a(n) = 20*A008854(n+1) - 5.
a(n) = 100*n/3 - 80*sin(2*n*Pi/3)/(3*sqrt(3)) - 5.
G.f.: (5*x*(x^3 + 4*x^2 + 12*x + 3))/((x^2 + x + 1)*(x - 1)^2).
E.g.f.: 100*x*exp(x)/3 - 80*sin(sqrt(3)*x/2)/(exp(x/2)*(3*sqrt(3)))-5*exp(x).
(End)

cp's OEIS Frontend

A019952 Decimal expansion of tangent of 54 degrees.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A090298 Permutation of natural numbers generated by 5-row array shown below.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A247643 a(n) = ( 10*n*(n+1)+(2*n+1)*(-1)^n+7 )/8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A063225 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 62 ).

Views

Author

Keywords

Links

Crossrefs

Formula

A317657 Numbers congruent to {15, 75, 95} mod 100.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Formula

A247643 a(n) = ( 10n(n+1)+(2n+1)(-1)^n+7 )/8.