A188593 -id:A188593 - 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

A182007 Decimal expansion of 2*sin(Pi/5).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 06 2012

The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).
Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.
Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014
This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016
rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.

			1.1755705045849462583374119...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pentagon.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A002193, A003401, A019881, A019845, A063226, A102769, A131595, A188593, A300074.

SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

evalf(2*sin(Pi/5),100); # Muniru A Asiru, Nov 02 2018

RealDigits[2*Sin[Pi/5],10,120][[1]] (* Harvey P. Dale, Sep 29 2012 *)

2*sin(Pi/5) \\ Stanislav Sykora, May 02 2016

Equals sqrt(3-phi).
Equals sqrt((5-sqrt(5))/2). - Jean-François Alcover, May 21 2013
Equals Product_{k>=0} ((10*k + 4)*(10*k + 6))/((10*k + 3)*(10*k + 7)). - Antonio Graciá Llorente, Mar 25 2024
Equals Product_{k>=1} (1 - (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A019845 = 1/A300074. - Hugo Pfoertner, Nov 23 2024

A005531 Decimal expansion of fifth root of 2.

Original entry on oeis.org

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

A188618 Decimal expansion of (diagonal)/(shortest side) of 1st electrum rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 06 2011

The 1st electrum rectangle is introduced here as a rectangle whose length L and width W satisfy L/W=(1+sqrt(3))/2.  The name of this shape refers to the alloy of gold and silver known as electrum, in view of the existing names "golden rectangle" and "silver rectangle" and these continued fractions:
golden ratio: L/W=[1,1,1,1,1,1,1,1,1,1,1,...]
silver ratio: L/W=[2,2,2,2,2,2,2,2,2,2,2,...]
1st electrum ratio: L/W=[1,2,1,2,1,2,1,2,...]
2nd electrum ratio: L/W=[2,1,2,1,2,1,2,1,...].
Recall that removal of 1 square from a golden rectangle leaves a golden rectangle, and that removal of 2 squares from a silver rectangle leaves a silver rectangle.  Removal of a square from a 1st electrum rectangle leaves a silver rectangle; removal of 2 squares from a 2nd electrum rectangle leaves a golden rectangle.

			1.6929339632083818...

Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Cf. A188593 (golden), A121601 (silver), A188619 (2nd electrum).

h=(1+3^(1/2))/2; (* continued fraction: h=[1,2,1,2,...]. *)
r=(1+h^2)^(1/2);
RealDigits[N[r, 130]][[1]]

Equals sqrt(2+(1/2)sqrt(3)).

A188619 Decimal expansion of (diagonal)/(shortest side) of 2nd electrum rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 06 2011

The 2nd electrum rectangle is introduced here as a rectangle whose length L and width W satisfy L/W=1+sqrt(3).  The name of this shape refers to the alloy of gold and silver known as electrum, in view of the existing names "golden rectangle" and "silver rectangle" and these continued fractions:
golden ratio: L/W=[1,1,1,1,1,1,1,1,1,1,1,...]
silver ratio: L/W=[2,2,2,2,2,2,2,2,2,2,2,...]
1st electrum ratio: L/W=[1,2,1,2,1,2,1,2,...]
2nd electrum ratio: L/W=[2,1,2,1,2,1,2,1,...].
Recall that removal of 1 square from a golden rectangle leaves a golden rectangle, and that removal of 2 squares from a silver rectangle leaves a silver rectangle.  Removal of a square from a 1st electrum rectangle leaves a silver rectangle; removal of 2 squares from a 2nd electrum rectangle leaves a golden rectangle.

			(diagonal/shortest side) = 2.9093129111764094646 approximately.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.13 Steinitz Constants, p. 241.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Cf. A188593 (golden), A121601 (silver), A188618 (1st electrum).

SetDefaultRealField(RealField(100)); Sqrt(5+2*Sqrt(3)); // G. C. Greubel, Nov 02 2018

h = 1 + 3^(1/2); r = (1 + h^2)^(1/2)
FullSimplify[r]
N[r, 130] (* ratio of diagonal h to shortest side; h=[1,2,1,2,1,2,...] *)
RealDigits[N[r, 130]][[1]]
RealDigits[Sqrt[5 + 2*Sqrt[3]], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)

default(realprecision, 100); sqrt(5+2*sqrt(3)) \\ G. C. Greubel, Nov 02 2018

Equals sqrt(5+2*sqrt(3)).

A158934 Decimal expansion of xi = (cos(Pi/5) - 1/2) / (sin(Pi/5) + 1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Mar 31 2009

This constant xi arises in the Davenport-Heilbronn zeta-function Z(s)=Sum_{k>=1} b(k)/k^s where b(k) is the 5-periodic sequence with period [1,xi,-xi,0]. Z satisfies a functional equation (like zeta) but does not satisfy RH. Some nontrivial zeros are off the critical line (see reference).

			0.2840790438404122960282...

Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller, The Riemann Hypothesis, Springer, 2009, pp. 136-137.

Bruce C. Berndt, Heng Huat Chan and Liang-Cheng Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, Journal für die reine und angewandte Mathematik, Vol. 480 (1996), pp. 141-160, eq. (1.1).
Harold Davenport and Hans Heilbronn, On the zeros of certain Dirichlet series, Journal of the London Mathematical Society, Vol. s1-11, No. 3 (1936), pp. 181-185.
Harold Davenport and Hans Heilbronn, On the zeros of certain Dirichlet series (Second paper), Journal of the London Mathematical Society, Vol. s1-11, No. 4 (1936), pp. 307-312.

Cf. A001622, A158241, A188593, A019845.

(Sqrt[5]-1) / (2+Sqrt[10-2*Sqrt[5]]) // RealDigits[#, 10, 104]& // First (* Jean-François Alcover, Mar 04 2013 *)

PARI
```
xi=(cos(Pi/5)-1/2)/(sin(Pi/5)+1/2)
```

Equals (sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1).
Equals (A001622-1)/(2*A019845+1). - R. J. Mathar, Apr 02 2009
Equals sqrt((5 + sqrt(5))/2) - (sqrt(5) + 1)/2 = A188593 - A001622. - Amiram Eldar, Jan 23 2022

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.

A358938 Decimal expansion of the real root of 2*x^5 - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Dec 07 2022

This is the reciprocal of A005531.

The other two complex conjugate pairs of roots are obtained, with the present number r = (1/2)^(1/5) and the golden section phi (A001622), from x1 = r*exp(Pi*i*2/5) = r*(phi - 1 + sqrt(2 + phi)*i)/2 = r*(A001622 - 1 + A188593*i)/2 = 0.2690149185... + 0.8279427859...*i, x2 = r*exp(Pi*i*4/5) = r*(-phi + sqrt(3 - phi)*i)/2 = r*(-A001622 + A182007*i)/2 = -0.7042902001... + 0.5116967824...*i.

			0.87055056329612413913627001747974609897912542434800304824185956850675...

Cf. A001622, A182007, A188593, A005531, A011101.

RealDigits[Surd[1/2, 5], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)

r = (1/2)^(1/5) = 1/A005531.

Equals A011101/2. - Hugo Pfoertner, Mar 24 2025

cp's OEIS Frontend

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

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A182007 Decimal expansion of 2*sin(Pi/5).

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A005531 Decimal expansion of fifth root of 2.

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A188618 Decimal expansion of (diagonal)/(shortest side) of 1st electrum rectangle.

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A188619 Decimal expansion of (diagonal)/(shortest side) of 2nd electrum rectangle.

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A158934 Decimal expansion of xi = (cos(Pi/5) - 1/2) / (sin(Pi/5) + 1/2).

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