A019970 -id:A019970 - cp's OEIS frontend

A090771 Numbers that are congruent to {1, 9} mod 10.

1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69, 71, 79, 81, 89, 91, 99, 101, 109, 111, 119, 121, 129, 131, 139, 141, 149, 151, 159, 161, 169, 171, 179, 181, 189, 191, 199, 201, 209, 211, 219, 221, 229, 231, 239, 241, 249, 251, 259, 261, 269, 271, 279, 281

Offset: 1

Giovanni Teofilatto, Feb 07 2004

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 10). - Bruno Berselli, Nov 17 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A000796, A090298, A005408, A047209, A007310, A019970, A047336, A047522, A091998, A094888, A113801, A175886, A175887, A188593.
Cf. A056020 (n = 1 or 8 mod 9), A175885 (n = 1 or 10 mod 11).
Cf. A045468 (primes), A195142 (partial sums).

a090771 n = a090771_list !! (n-1)
a090771_list = 1 : 9 : map (+ 10) a090771_list
-- Reinhard Zumkeller, Jan 07 2012

Flatten[Table[10n - {9, 1}, {n, 30}]] (* Alonso del Arte, Sep 02 2014 *)
LinearRecurrence[{1,1,-1},{1,9,11},60] (* Harvey P. Dale, Jul 05 2020 *)

a(n)=n\2*10-(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = sqrt(40*A057569(n) + 1). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Sep 16 2010 - Nov 17 2010: (Start)
  G.f.: x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).
  a(n) = (10*n + 3*(-1)^n - 5)/2.
  a(n) = -a(-n + 1) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 10.
  a(n) = 10*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
a(n) = 10*n - a(n-1) - 10 (with a(1) = 1). - Vincenzo Librandi, Nov 16 2010
a(n) = sqrt(10*A132356(n-1) + 1). - Ivan N. Ianakiev, Nov 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/10)*cot(Pi/10) = A000796 * A019970 / 10 = sqrt(5 + 2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((10*x - 5)*exp(x) + 3*exp(-x))/2. - David Lovler, Sep 03 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(phi+2) (A188593).
Product_{n>=2} (1 + (-1)^n/a(n)) = Pi*phi/5 = A094888/10. (End)

Edited and extended by Ray Chandler, Feb 10 2004

A019887 Decimal expansion of sine of 78 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Equals sin(13*Pi/30). - Wesley Ivan Hurt, Aug 31 2014

A quartic number with denominator 2 and minimal polynomial 16x^4 + 8x^3 - 16x^2 - 8x + 1. - Charles R Greathouse IV, Aug 27 2017

			0.9781476007338056379285667478695995324597378088626771078851...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4

Digits:=100: evalf(sin(13*Pi/30)); # Wesley Ivan Hurt, Aug 31 2014

First@RealDigits[Sin[78 Degree], 10, 120] (* Michael De Vlieger, Aug 31 2014 *)

cos(Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/15) = [sqrt(5)-1]*[1+sqrt(3)*sqrt{5+2*sqrt(5)}]/8 = [A002163-1]*[1+A002194*A019970]/8. - R. J. Mathar, Jun 18 2006
Equals 2*A019848*A019860. - R. J. Mathar, Jan 17 2021
4*this^3 -3*this = A019863. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/10,1/10 ; 1/2 ; 3/4). - R. J. Mathar, Aug 31 2025
A root of 16*x^4+8*x^3-16*x^2-8*x+1=0. - R. J. Mathar, Aug 31 2025

A019934 Decimal expansion of tangent of 36 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 54 degrees. - Mohammad K. Azarian, Jun 30 2013

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			0.72654252800536088589546675748061874961609239296520...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 4.

Cf. A019934, A019952, A019970, A002163, A063226.

RealDigits[Tan[36 Degree],10,120][[1]] (* Harvey P. Dale, Nov 06 2012 *)

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

This number is sqrt(5-2*sqrt(5)). This number * A019970 = sqrt(5) = A002163. - R. J. Mathar, Jun 18 2006
The smallest positive solution of cos(4*arctan(x)) = cos(6*arctan(x)). - Thomas Olson, Oct 03 2014
Let r(n) = (n - 1)/(n + 1) if n mod 4 = 1, (n + 1)/(n - 1) otherwise; then this constant (A019934) equals with Product_{n>=0} r(10*n+5) = (2/3) * (8/7) * (12/13) * (18/17) * ... - Dimitris Valianatos, Sep 14 2019
Equals Product_{k>=1} (1 + (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 1/A019952. - Hugo Pfoertner, Nov 23 2024
tan(Pi/5) = A019845 / A019863. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of x^4-10*x^2+5=0. (Other A019970). - R. J. Mathar, Aug 31 2025

A019916 Decimal expansion of tan(Pi/10) (angle of 18 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

In a regular pentagon inscribed in a unit circle this is the cube of the length of the side divided by 5: (1/5)*(sqrt(3 - phi))^3 with phi from A001622. - Wolfdieter Lang, Jan 08 2018

Quartic number of denominator 5 and minimal polynomial 5x^4 - 10x^2 + 1. - Charles R Greathouse IV, May 13 2019

			0.3249196962329063261558714122151344649549034715214751003078047191...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827 (sin(Pi/10)), A019881 (cos(Pi/10)).

RealDigits[Tan[18 Degree],10,120][[1]] (* Harvey P. Dale, Mar 07 2012 *)

tan(Pi/10) \\ Michel Marcus, Jan 08 2018

polrootsreal(5*x^4-10*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals A019827/A019881 = 1/A019970 = 1/sqrt(5+2*sqrt(5)). - R. J. Mathar, Jul 26 2010
Equals tan((phi - 1)/sqrt(2 + phi)) = (1/5)*(sqrt(3 - phi))^3 = (3 - phi)*sqrt(3 - phi)/5 = sqrt(7 - 4*phi)/(2*phi - 1), with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Equals Product_{k>=0} ((5*k + 1)/(5*k + 4))^(-1)^(k) = Product_{k>=0} A090771(k)/A090773(k). - Antonio Graciá Llorente, Mar 24 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

A377796 Decimal expansion of the surface area of a truncated icosidodecahedron (great rhombicosidodecahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Nov 07 2024

			174.292030342323920882932107526283465728485221920...

Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron.
Wikipedia, Truncated icosidodecahedron.

Cf. A377797 (volume), A377798 (circumradius), A377799 (midradius).

Cf. A090388, A019970.

First[RealDigits[30*(1 + Sqrt[3] + Sqrt[5 + Sqrt[20]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosidodecahedron", "SurfaceArea"], 10, 100]]

Equals 30*(1 + sqrt(3) + sqrt(5 + 2*sqrt(5))) = 30*(A090388 + A019970).

A019982 Decimal expansion of tangent of 84 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 6 degrees. - Ivan Panchenko, Sep 01 2014

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Exact trigonometric constants.

Cf. A019970, A019976.

RealDigits[Tan[7*Pi/15], 10, 100][[1]] (* Amiram Eldar, Apr 07 2022 *)

From Amiram Eldar, Apr 07 2022: (Start)
Equals tan(78 degrees) + tan(72 degrees) + tan(60 degrees).
Equals sqrt(23 + 10*sqrt(5) + 2*sqrt(3*(85 + 38*sqrt(5)))).
The 4th positive and largest root of x^8 - 92*x^6 + 134*x^4 - 28*x^2 + 1 = 0. (End)

A309434 a(n) = floor(nIm(2e^(iPi/5))/(Im(2e^(i*Pi/5)) - 1)).

Original entry on oeis.org

6, 13, 20, 26, 33, 40, 46, 53, 60, 66, 73, 80, 87, 93, 100, 107, 113, 120, 127, 133, 140, 147, 154, 160, 167, 174, 180, 187, 194, 200, 207, 214, 220, 227, 234, 241, 247, 254, 261, 267, 274, 281, 287, 294, 301, 308, 314, 321, 328, 334, 341

Offset: 1

Karl V. Keller, Jr., Jun 06 2020

nonn

This is the Beatty sequence for Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1).
This is the complement of A335137.
Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1) = (5 + sqrt(5))/2 + sqrt(5 + 2*sqrt(5)) = 6.695717525925148250774877410... = 2 + phi + tan(2*Pi/5) = A296184 + A019970.
For n < 10, a(n) = A109235(n).
Re(2*e^(i*Pi/5))/(Re(2*e^(i*Pi/5)) - 1) = (3 + sqrt(5))/2 = 1 + phi = phi^2 = A104457.
Floor(n*Re(2*e^(i*Pi/5))/(Re(2*e^(i*Pi/5)) - 1)) is A001950 (floor(n*phi^2)).

			For n = 3, floor(3*6.69571) = 20.

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. A001950, A019970, A104457, A109235, A296184, A335137.

a[n_] := Floor[n * Im[2 * Exp[I * Pi/5]]/(Im[2 * Exp[I * Pi/5]] - 1)]; Array[a, 100] (* Amiram Eldar, Jul 06 2020 *)

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

from sympy import floor, sqrt
for n in range(1, 101): print(floor(n*((5 + sqrt(5))/2 + sqrt(5 + 2*sqrt(5)))), end=', ')

A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2

Offset: 4

Mohammed Yaseen, Aug 22 2020

			Triangle begins:
1;
2, 2;
2, 2, 2;
2, 2, 2, 2;
2, 2, 3, 2, 2;
2, 3, 3, 3, 3, 2;
2, 3, 3, 3, 3, 3, 2;
2, 3, 3, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 4, 3, 3, 2;
2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2;
...
Row n lists the closest integers to the length of the diagonals drawn from a fixed vertex of a regular n-gon with unit edge length, n >= 4.
The lengths of the diagonals drawn from vertex A of a regular 8-gon ABCDEFGH with unit edge length are:
AC = 1.84775...
AD = 2.41421...
AE = 2.61312...
AF = 2.41421...
AG = 1.84775...
So the row for n=8 is 2, 2, 3, 2, 2.

Cf. A064313.
Decimal expansion of diagonal lengths of regular n-gons with unit edge length:
n=4  A002193.
n=5  A001622.
n=6  A002194, A000038.
n=7  A160389, A231187.
n=8  A179260, A014176, A121601.
n=9  A332437.
n=10 A188593, A104457, A019970, A134945.
n=11 A231186.
n=12 A188887, A090388, A337402, A019973, A214726.

T[n_,k_]:=Round[Sin[(k+1)*Pi/n]/Sin[Pi/n]]; Flatten[Table[T[n,k],{n,4,16},{k,1,n-3}]] (* Stefano Spezia, Sep 07 2020 *)

T(n,k) = round(sin((k+1)*Pi/n)/sin(Pi/n)), n >= 4, 1 <= k <= n-3.

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.

cp's OEIS Frontend

A090771 Numbers that are congruent to {1, 9} mod 10.

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A019887 Decimal expansion of sine of 78 degrees.

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A019934 Decimal expansion of tangent of 36 degrees.

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A019916 Decimal expansion of tan(Pi/10) (angle of 18 degrees).

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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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A377796 Decimal expansion of the surface area of a truncated icosidodecahedron (great rhombicosidodecahedron) with unit edge length.

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A019982 Decimal expansion of tangent of 84 degrees.

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

A309434 a(n) = floor(nIm(2e^(iPi/5))/(Im(2e^(i*Pi/5)) - 1)).