A102208 -id:A102208 - cp's OEIS frontend

A001622 Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.

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

Offset: 1

N. J. A. Sloane

Also decimal expansion of the positive root of (x+1)^n - x^(2n). (x+1)^n - x^(2n) = 0 has only two real roots x1 = -(sqrt(5)-1)/2 and x2 = (sqrt(5)+1)/2 for all n > 0. - Cino Hilliard, May 27 2004
The golden ratio phi is the most irrational among irrational numbers; its successive continued fraction convergents F(n+1)/F(n) are the slowest to approximate to its actual value (I. Stewart, in "Nature's Numbers", Basic Books, 1997). - Lekraj Beedassy, Jan 21 2005
Let t=golden ratio. The lesser sqrt(5)-contraction rectangle has shape t-1, and the greater sqrt(5)-contraction rectangle has shape t. For definitions of shape and contraction rectangles, see A188739. - Clark Kimberling, Apr 16 2011
The golden ratio (often denoted by phi or tau) is the shape (i.e., length/width) of the golden rectangle, which has the special property that removal of a square from one end leaves a rectangle of the same shape as the original rectangle. Analogously, removals of certain isosceles triangles characterize side-golden and angle-golden triangles. Repeated removals in these configurations result in infinite partitions of golden rectangles and triangles into squares or isosceles triangles so as to match the continued fraction, [1,1,1,1,1,...] of tau. For the special shape of rectangle which partitions into golden rectangles so as to match the continued fraction [tau, tau, tau, ...], see A188635. For other rectangular shapes which depend on tau, see A189970, A190177, A190179, A180182. For triangular shapes which depend on tau, see A152149 and A188594; for tetrahedral, see A178988. - Clark Kimberling, May 06 2011
Given a pentagon ABCDE, 1/(phi)^2 <= (A*C^2 + C*E^2 + E*B^2 + B*D^2 + D*A^2) / (A*B^2 + B*C^2 + C*D^2 + D*E^2 + E*A^2) <= (phi)^2. - Seiichi Kirikami, Aug 18 2011
If a triangle has sides whose lengths form a geometric progression in the ratio of 1:r:r^2 then the triangle inequality condition requires that r be in the range 1/phi < r < phi. - Frank M Jackson, Oct 12 2011
The graphs of x-y=1 and x*y=1 meet at (tau,1/tau). - Clark Kimberling, Oct 19 2011
Also decimal expansion of the first root of x^sqrt(x+1) = sqrt(x+1)^x. - Michel Lagneau, Dec 02 2011
Also decimal expansion of the root of (1/x)^(1/sqrt(x+1)) = (1/sqrt(x+1))^(1/x). - Michel Lagneau, Apr 17 2012
This is the case n=5 of (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)): (1+sqrt(5))/2 = (Gamma(1/5)/Gamma(3/5))*(Gamma(4/5)/Gamma(2/5)). - Bruno Berselli, Dec 14 2012
Also decimal expansion of the only number x>1 such that (x^x)^(x^x) = (x^(x^x))^x = x^((x^x)^x). - Jaroslav Krizek, Feb 01 2014
For n >= 1, round(phi^prime(n)) == 1 (mod prime(n)) and, for n >= 3, round(phi^prime(n)) == 1 (mod 2*prime(n)). - Vladimir Shevelev, Mar 21 2014
The continuous radical sqrt(1+sqrt(1+sqrt(1+...))) tends to phi. - Giovanni Zedda, Jun 22 2019
Equals sqrt(2+sqrt(2-sqrt(2+sqrt(2-...)))). - Diego Rattaggi, Apr 17 2021
Given any complex p such that real(p) > -1, phi is the only real solution of the equation z^p+z^(p+1)=z^(p+2), and the only attractor of the complex mapping z->M(z,p), where M(z,p)=(z^p+z^(p+1))^(1/(p+2)), convergent from any complex plane point. - Stanislav Sykora, Oct 14 2021
The only positive number such that its decimal part, its integral part and the number itself (x-[x], [x] and x) form a geometric progression is phi, with respectively (phi -1, 1, phi) and a ratio = phi. This is the answer to the 4th problem of the 7th Canadian Mathematical Olympiad in 1975 (see IMO link and Doob reference). - Bernard Schott, Dec 08 2021
The golden ratio is the unique number x such that f(n*x)*c(n/x) - f(n/x)*c(n*x) = n for all n >= 1, where f = floor and c = ceiling. - Clark Kimberling, Jan 04 2022
In The Second Scientific American Book Of Mathematical Puzzles and Diversions, Martin Gardner wrote that, by 1910, Mark Barr (1871-1950) gave phi as a symbol for the golden ratio. - Bernard Schott, May 01 2022
Phi is the length of the equal legs of an isosceles triangle with side c = phi^2, and internal angles (A,B) = 36 degrees, C = 108 degrees. - Gary W. Adamson, Jun 20 2022
The positive solution to x^2 - x - 1 = 0. - Michal Paulovic, Jan 16 2023
The minimal polynomial of phi^n, for nonvanishing integer n, is P(n, x) = x^2 - L(n)*x + (-1)^n, with the Lucas numbers L = A000032, extended to negative arguments with L(n) = (-1)^n*L(n). P(0, x) = (x - 1)^2 is not minimal. - Wolfdieter Lang, Feb 20 2025
This is the largest real zero x of (x^4 + x^2 + 1)^2 = 2*(x^8 + x^4 + 1). - Thomas Ordowski, May 14 2025

			1.6180339887498948482045868343656381177203091798057628621...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 112, 123, 184, 190, 203.
Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993 - Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1975, pages 76-77, 1993.
Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific, River Edge, NJ, 1997.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, Section 1.2.
Martin Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, Simon & Schuster, NY, 1961.
Martin Gardner, Weird Water and Fuzzy Logic: More Notes of a Fringe Watcher, "The Cult of the Golden Ratio", Chapter 9, Prometheus Books, 1996, pages 90-97.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.5 The Fibonacci and Related Sequences, p. 287.
H. E. Huntley, The Divine Proportion, Dover, NY, 1970.
Mario Livio, The Golden Ratio, Broadway Books, NY, 2002. [see the review by G. Markowsky in the links field]
Gary B. Meisner, The Golden Ratio: The Divine Beauty of Mathematics, Race Point Publishing (The Quarto Group), 2018. German translation: Der Goldene Schnitt, Librero, 2023.
Scott Olsen, The Golden Section, Walker & Co., NY, 2006.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 137-139.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Hans Walser, The Golden Section, Math. Assoc. of Amer. Washington DC 2001.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 36-40.
Claude-Jacques Willard, Le nombre d'or, Magnard, Paris, 1987.

Robert G. Wilson v, Table of n, a(n) for n = 1..100000
Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176; Solution, ibid., Vol. 12, No. 1 (Winter 2000), pp. 61-62.
John Baez, This week's finds in mathematical physics, Week 203.
John Baez, The Rankin Lectures 2008, My Favorite Numbers: 5. [video]
Murray Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fib. Quart., Vol. 4, No. 2 (1961), pp. 157-162.
Ömür Deveci, Zafer Adıgüzel and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
T. Eveilleau, Le nombre d'or (in French).
Abdul Gaffar, Anand B. Joshi, Sonali Singh, and Keerti Srivastava, A high capacity multi-image steganography technique based on golden ratio and non-subsampled contourlet transform, Multimedia Tools and Applications (2022).
Gutenberg Project, The golden ratio to 20000 places.
ICON Project, The golden ratio to 50000 places.
The IMO Compendium, Problem 4, 7th Canadian Mathematical Olympiad 1975.
L. B. W. Jolley, Summation of Series, Dover, 1961
Franklin H. J. Kenter, It's good to be phi: a solution to a problem of Gosper and Knuth, arXiv:1712.04856 [math.HO], 2017.
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, Vol. 11 (2007), pp. 165-171.
Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
Ron Knott, Fibonacci numbers and the golden section.
Wolfdieter Lang, A list of representative simple difference sets of the Singer type for small orders m, Karlsruher Institut für Technologie (Karlsruhe, Germany 2020).
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Simon Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, Vol. 1, No. 4 (2005), pp. 499-512.
Gary B. Meisner, Phi, The Golden Number.
George Markowsky, Misconceptions About the Golden Ratio, College Mathematics Journal, 23:1 (January 1992), 2-19.
George Markowsky, Book review: The Golden Ratio, Notices of the AMS, 52:3 (March 2005), 344-347.
R. S. Melham and A. G. Shannon, Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
Jean-Christophe Michel, Le nombre d'or.
J. J. O'Connor and E. F. Robertson, The Golden ratio.
Hugo Pfoertner, 1 million digits of phi, Computed using A. J. Yee's y-cruncher.
Simon Plouffe, Plouffe's Inverter, The golden ratio to 10 million digits. [Only announcement, file truncated]
Simon Plouffe, The golden ratio:(1+sqrt(5))/2 to 20000 places.
Fred Richman, Fibonacci sequence with multiprecision Java, Successive approximations to phi from ratios of consecutive Fibonacci numbers.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4, No. 35 (Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.
E. F. Schubert, The Fibonacci series.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014).
Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, Vol. 1385, pp. 97-100; arXiv:1106.4246 [math.NT], 2011.
Matthew R. Watkins, The "Golden Mean" in number theory.
Eric Weisstein's World of Mathematics, Golden Ratio.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Silver Ratio.
Wikipedia, Mark Barr.
Wikipedia, Golden ratio.
Wikipedia, Kronecker-Weber theorem.
Wikipedia, Metallic mean.
Alexander J. Yee, y-cruncher - A Multi-Threaded Pi-Program.
Index entries for algebraic numbers, degree 2.
Index to sequences related to Olympiads.

Cf. A000012 (continued fraction coefficients), A000032, A000045, A006497, A080039, A104457, A188635, A192222, A192223, A145996, A139339, A197762, A002163, A094874, A134973.
Cf. A102208, A102769, A131595.
Cf. A302973, A303069, A304022.

Digits:=1000; evalf((1+sqrt(5))/2); # Wesley Ivan Hurt, Nov 01 2013

RealDigits[(1 + Sqrt[5])/2, 10, 130] (* Stefan Steinerberger, Apr 02 2006 *)
RealDigits[ Exp[ ArcSinh[1/2]], 10, 111][[1]] (* Robert G. Wilson v, Mar 01 2008 *)
RealDigits[GoldenRatio,10,120][[1]] (* Harvey P. Dale, Oct 28 2015 *)

default(realprecision, 20080); x=(1+sqrt(5))/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b001622.txt", n, " ", d));  \\ Harry J. Smith, Apr 19 2009

/* Digit-by-digit method: write it as 0.5+sqrt(1.25) and start at hundredths digit */
r=11; x=400; print(1); print(6);
for(dig=1, 110, {d=0; while((20*r+d)*d <= x, d++);
d--; /* while loop overshoots correct digit */
print(d); x=100*(x-(20*r+d)*d); r=10*r+d})
\\ Michael B. Porter, Oct 24 2009

a(n) = floor(10^(n-1)*(quadgen(5))%10);
alist(len) = digits(floor(quadgen(5)*10^(len-1))); \\ Chittaranjan Pardeshi, Jun 22 2022

from sympy import S
def alst(n): # truncate extra last digit to avoid rounding
  return list(map(int, str(S.GoldenRatio.n(n+1)).replace(".", "")))[:-1]
print(alst(105)) # Michael S. Branicky, Jan 06 2021

Equals Sum_{n>=2} 1/A064170(n) = 1/1 + 1/2 + 1/(2*5) + 1/(5*13) + 1/(13*34) + ... - Gary W. Adamson, Dec 15 2007
Equals Hypergeometric2F1([1/5, 4/5], [1/2], 3/4) = 2*cos((3/5)*arcsin(sqrt(3/4))). - Artur Jasinski, Oct 26 2008
From Hieronymus Fischer, Jan 02 2009: (Start)
The fractional part of phi^n equals phi^(-n), if n is odd. For even n, the fractional part of phi^n is equal to 1-phi^(-n).
General formula: Provided x>1 satisfies x-x^(-1)=floor(x), where x=phi for this sequence, then:
for odd n: x^n - x^(-n) = floor(x^n), hence fract(x^n) = x^(-n),
for even n: x^n + x^(-n) = ceiling(x^n), hence fract(x^n) = 1 - x^(-n),
for all n>0: x^n + (-x)^(-n) = round(x^n).
x=phi is the minimal solution to x - x^(-1) = floor(x) (where floor(x)=1 in this case).
Other examples of constants x satisfying the relation x - x^(-1) = floor(x) include A014176 (the silver ratio: where floor(x)=2) and A098316 (the "bronze" ratio: where floor(x)=3). (End)
Equals 2*cos(Pi/5) = e^(i*Pi/5) + e^(-i*Pi/5). - Eric Desbiaux, Mar 19 2010
The solutions to x-x^(-1)=floor(x) are determined by x=(1/2)*(m+sqrt(m^2+4)), m>=1; x=phi for m=1. In terms of continued fractions the solutions can be described by x=[m;m,m,m,...], where m=1 for x=phi, and m=2 for the silver ratio A014176, and m=3 for the bronze ratio A098316. - Hieronymus Fischer, Oct 20 2010
Sum_{n>=1} x^n/n^2 = Pi^2/10 - (log(2)*sin(Pi/10))^2 where x = 2*sin(Pi/10) = this constant here. [Jolley, eq 360d]
phi = 1 + Sum_{k>=1} (-1)^(k-1)/(F(k)*F(k+1)), where F(n) is the n-th Fibonacci number (A000045). Proof. By Catalan's identity, F^2(n) - F(n-1)*F(n+1) = (-1)^(n-1). Therefore,(-1)^(n-1)/(F(n)*F(n+1)) = F(n)/F(n+1) - F(n-1)/F(n). Thus Sum_{k=1..n} (-1)^(k-1)/(F(k)*F(k+1)) = F(n)/F(n+1). If n goes to infinity, this tends to 1/phi = phi - 1. - Vladimir Shevelev, Feb 22 2013
phi^n = (A000032(n) + A000045(n)*sqrt(5)) / 2. - Thomas Ordowski, Jun 09 2013
Let P(q) = Product_{k>=1} (1 + q^(2*k-1)) (the g.f. of A000700), then A001622 = exp(Pi/6) * P(exp(-5*Pi)) / P(exp(-Pi)). - Stephen Beathard, Oct 06 2013
phi = i^(2/5) + i^(-2/5) = ((i^(4/5))+1) / (i^(2/5)) = 2*(i^(2/5) - (sin(Pi/5))i) = 2*(i^(-2/5) + (sin(Pi/5))i). - Jaroslav Krizek, Feb 03 2014
phi = sqrt(2/(3 - sqrt(5))) = sqrt(2)/A094883. This follows from the fact that ((1 + sqrt(5))^2)*(3 - sqrt(5)) = 8, so that ((1 + sqrt(5))/2)^2 = 2/(3 - sqrt(5)). - Geoffrey Caveney, Apr 19 2014
exp(arcsinh(cos(Pi/2-log(phi)*i))) = exp(arcsinh(sin(log(phi)*i))) = (sqrt(3) + i) / 2. - Geoffrey Caveney, Apr 23 2014
exp(arcsinh(cos(Pi/3))) = phi. - Geoffrey Caveney, Apr 23 2014
cos(Pi/3) + sqrt(1 + cos(Pi/3)^2). - Geoffrey Caveney, Apr 23 2014
2*phi = z^0 + z^1 - z^2 - z^3 + z^4, where z = exp(2*Pi*i/5). See the Wikipedia Kronecker-Weber theorem link. - Jonathan Sondow, Apr 24 2014
phi = 1/2 + sqrt(1 + (1/2)^2). - Geoffrey Caveney, Apr 25 2014
Phi is the limiting value of the iteration of x -> sqrt(1+x) on initial value a >= -1. - Chayim Lowen, Aug 30 2015
From Isaac Saffold, Feb 28 2018: (Start)
1 = Sum_{k=0..n} binomial(n, k) / phi^(n+k) for all nonnegative integers n.
1 = Sum_{n>=1} 1 / phi^(2n-1).
1 = Sum_{n>=2} 1 / phi^n.
phi = Sum_{n>=1} 1/phi^n. (End)
From Christian Katzmann, Mar 19 2018: (Start)
phi = Sum_{n>=0} (15*(2*n)! + 8*n!^2)/(2*n!^2*3^(2*n+2)).
phi = 1/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
phi = Product_{k>=1} (1 + 2/(-1 + 2^k*(sqrt(4+(1-2/2^k)^2) + sqrt(4+(1-1/2^k)^2)))). - Gleb Koloskov, Jul 14 2021
Equals Product_{k>=1} (Fibonacci(3*k)^2 + (-1)^(k+1))/(Fibonacci(3*k)^2 + (-1)^k) (Melham and Shannon, 1995). - Amiram Eldar, Jan 15 2022
From Michal Paulovic, Jan 16 2023: (Start)
Equals the real part of 2 * e^(i * Pi / 5).
Equals 2 * sin(3 * Pi / 10) = 2*A019863.
Equals -2 * sin(37 * Pi / 10).
Equals 1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / ...)))).
Equals (2 + 3 * (2 + 3 * (2 + 3 * ...)^(1/4))^(1/4))^(1/4).
Equals (1 + 2 * (1 + 2 * (1 + 2 * ...)^(1/3))^(1/3))^(1/3).
Equals (1 + phi + (1 + phi + (1 + phi + ...)^(1/3))^(1/3))^(1/3).
Equals 13/8 + Sum_{k=0..oo} (-1)^(k+1)*(2*k+1)!/((k+2)!*k!*4^(2*k+3)).
(End)
phi^n = phi * A000045(n) + A000045(n-1). - Gary W. Adamson, Sep 09 2023
The previous formula holds for integer n, with F(-n) = (-1)^(n+1)*F(n), for n >= 0, with F(n) = A000045(n), for n >= 0. phi^n are integers in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Sep 16 2023
Equals Product_{k>=0} ((5*k + 2)*(5*k + 3))/((5*k + 1)*(5*k + 4)). - Antonio Graciá Llorente, Feb 24 2024
From Antonio Graciá Llorente, Apr 21 2024: (Start)
Equals Product_{k>=1} phi^(-2^k) + 1, with phi = A001622.
Equals Product_{k>=0} ((5^(k+1) + 1)*(5^(k-1/2) + 1))/((5^k + 1)*(5^(k+1/2) + 1)).
Equals Product_{k>=1} 1 - (4*(-1)^k)/(10*k - 5 + (-1)^k) = Product_{k>=1} A047221(k)/A047209(k).
Equals Product_{k>=0} ((5*k + 7)*(5*k + 1 + (-1)^k))/((5*k + 1)*(5*k + 7 + (-1)^k)).
Equals Product_{k>=0} ((10*k + 3)*(10*k + 5)*(10*k + 8)^2)/((10*k + 2)*(10*k + 4)*(10*k + 9)^2).
Equals Product_{k>=5} 1 + 1/(Fibonacci(k) - (-1)^k).
Equals Product_{k>=2} 1 + 1/Fibonacci(2*k).
Equals Product_{k>=2} (Lucas(k)^2 + (-1)^k)/(Lucas(k)^2 - 4*(-1)^k). (End)

Additional links contributed by Lekraj Beedassy, Dec 23 2003
More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 24 2004
More terms from Stefan Steinerberger, Apr 02 2006
Broken URL to Project Gutenberg replaced by Georg Fischer, Jan 03 2009
Edited by M. F. Hasler, Feb 24 2014

A019881 Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Circumradius of pentagonal pyramid (Johnson solid 2) with edge 1. - Vladimir Joseph Stephan Orlovsky, Jul 19 2010
Circumscribed sphere radius for a regular icosahedron with unit edges. - Stanislav Sykora, Feb 10 2014
Side length of the particular golden rhombus with diagonals 1 and phi (A001622); area is phi/2 (A019863). Thus, also the ratio side/(shorter diagonal) for any golden rhombus. Interior angles of a golden rhombus are always A105199 and A137218. - Rick L. Shepherd, Apr 10 2017
An algebraic number of degree 4; minimal polynomial is 16x^4 - 20x^2 + 5, which has these smaller, other solutions (conjugates): -A019881 < -A019845 < A019845 (sine of 36 degrees). - Rick L. Shepherd, Apr 11 2017
This is length ratio of one half of any diagonal in the regular pentagon and the circumscribing radius. - Wolfdieter Lang, Jan 07 2018
Quartic number of denominator 2 and minimal polynomial 16x^4 - 20x^2 + 5. - Charles R Greathouse IV, May 13 2019
This gives the imaginary part of one of the members of a conjugate pair of roots of x^5 - 1, with real part (-1 + phi)/2 = A019827, where phi = A001622. A member of the other conjugte pair of roots is (-phi + sqrt(3 - phi)*i)/2 = (-A001622 + A182007*i)/2 = -A001622/2  + A019845*i. - Wolfdieter Lang, Aug 30 2022

			0.95105651629515357211643933337938214340569863412575022244730564443015317008...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Golden Rhombus.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Wolfram Alpha, Johnson solid 2.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827, A102208, A179290 (inverse), A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A182007.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A179296 (dodecahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

SetDefaultRealField(RealField(100)); Sqrt((5 + Sqrt(5))/8); // G. C. Greubel, Nov 02 2018

Digits:=100: evalf(sin(2*Pi/5)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sqrt[(5 + Sqrt[5])/8], 10, 111]  (* Robert G. Wilson v *)
RealDigits[Sin[2 Pi/5], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

default(realprecision, 120);
real(I^(1/5)) \\ Rick L. Shepherd, Apr 10 2017

Equals sqrt((5+sqrt(5))/8) = cos(Pi/10). - Zak Seidov, Nov 18 2006
Equals 2F1(13/20,7/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals the real part of i^(1/5). - Stanislav Sykora, Apr 25 2012
Equals A001622*A182007/2. - Stanislav Sykora, Feb 10 2014
Equals sin(2*Pi/5) = sqrt(2 + phi)/2 = -sin(3*Pi/5), with phi = A001622  - Wolfdieter Lang, Jan 07 2018
Equals 2*A019845*A019863. - R. J. Mathar, Jan 17 2021

A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Regular icosahedron: A three-dimensional figure with 20 congruent equilateral triangle faces, 12 vertices, and 30 edges.
Shorter diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018
The length of the shorter side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022
The side length of a square inscribed within a golden ellipse with a unit semi-major axis. - Amiram Eldar, Oct 02 2022
(10/3)*(this constant)=3.504874080794224... is the volume of the polyhedron with 32 edges with conjectured maximum volume inscribed in a sphere of radius 1. It has 60 congruent triangular faces and the symmetry group of the regular icosahedron. See Pfoertner links for visualizations. - Hugo Pfoertner, Aug 02 2025

			1.051462224238267212051338169695753214570995864486683563057871046482422...

Muniru A Asiru, Table of n, a(n) for n = 1..1000
J. Brandts, S. Korotov, M. Krizek, and J. Solc, On nonobtuse simplicial partitions, Siam Rev. 51 (2) (2009) 317-335.
Dr. Math, Regular Polyhedra: Formulas.
Hugo Pfoertner, Visualization of the 32-edge polyhedron with conjecturally maximum volume, (2021).
Hugo Pfoertner, Number of edges incident with the vertices of the 32-edge polyhedron, video (2021).
Eric Weisstein's World of Mathematics, Golden Rhombus.
Eric Weisstein's World of Mathematics, Icosahedron.
Wikipedia, Icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A010527, A019881, A090773, A102208.

Cf. A179290 (longer golden rhombus diagonal).

evalf[120](csc(2*Pi/5)); # Muniru A Asiru, Dec 11 2018

RealDigits[Csc[2 Pi/5], 10, 110][[1]] (* Eric W. Weisstein, Dec 11 2018 *)

sqrt(50-10*sqrt(5))/5 \\ Charles R Greathouse IV, Jan 22 2024

from decimal import *
getcontext().prec = 110
c = Decimal.sqrt(2 - 2 / Decimal.sqrt(Decimal(5)))
print([int(i) for i in str(c) if i != '.'])
# Karl V. Keller, Jr., Jul 10 2020

Equals sqrt(50-10*sqrt(5))/5.
Equals csc(2*Pi/5). - Eric W. Weisstein, Dec 11 2018
Equals 1/Im(e^(3*i*Pi/5)) = 1/Im(e^(3*i*Pi/5) - 1) = sqrt(2 - 2/sqrt(5)). - Karl V. Keller, Jr., Jun 11 2020
Equals 1/A019881. - R. J. Mathar, Jan 17 2021
From Antonio Graciá Llorente, Mar 15 2024: (Start)
Equals Product_{k >= 1} ((10*k - 1)*(10*k + 1))/((10*k - 2)*(10*k + 2)).
Equals Product_{k >= 1} 1/(1 - 1/(25*(2*k - 1)^2)). (End)
Equals Product_{k>=1} (1 - (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 2024
A root of 5*x^4 - 20*x^2 + 16=0 (see A121570). - R. J. Mathar, Aug 29 2025

Partially rewritten by Charles R Greathouse IV, Feb 02 2011

A179294 Decimal expansion of radius of inscribed sphere about a regular icosahedron with edge = 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Icosahedron: A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.

			0.75576131407617073048013370202500139263844478889356105922958289203910...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Dr. Math, Icosahedron.
Wikipedia, Icosahedron.
MathWorld, Icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A010527, A102208, A179290, A179292.

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A237603 (dodecahedron).

RealDigits[(Sqrt[42+18Sqrt[5]]/12), 10, 175][[1]]

sqrt((7+3*sqrt(5))/6)/2 \\ Stefano Spezia, Jan 27 2025

Equals sqrt(42 + 18*sqrt(5))/12.

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A179587 Decimal expansion of the volume of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.
Also, decimal expansion of 1 + Product_{n>0} (1-1/(4*n+2)^2). - Bruno Berselli, Apr 02 2013
Decimal expansion of 1 + (least possible ratio of the side length of one inscribed square to the side length of another inscribed square in the same non-obtuse triangle). - L. Edson Jeffery, Nov 12 2014
2*sqrt(2)/3 is the radius of the base of the maximum-volume right cone inscribed in a unit-radius sphere. - Amiram Eldar, Sep 25 2022

			1.942809041582063365867792482806465385713114583584632048784453158660...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Victor Oxman and Moshe Stupel, Why are the side lengths of the squares inscribed in a triangle so close to each other?, Forum Geometricorum, Vol. 13 (2013), 113-115.
Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 2

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A224268.

Cf. A131594 (decimal expansion of sqrt(2)/3).

RealDigits[N[1+(2*Sqrt[2])/3,200]]
(* From the second comment: *) RealDigits[N[1 + Product[1 - 1/(4 n + 2)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

sqrt(8)/3+1 \\ Charles R Greathouse IV, Nov 14 2016

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

Equals 1 + 2*A131594. - L. Edson Jeffery, Nov 12 2014

A179292 Decimal expansion of radius of inscribed sphere of an icosahedron with radius of circumscribed sphere = 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Icosahedron: A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.

			0.794654472291766122955530928327594042026590588309264801754955775008438...

Mathforum, Icosahedron
Wikipedia, Icosahedron
MathWorld, Icosahedron
Index entries for algebraic numbers, degree 4.

Cf. A010527, A102208, A179290.

R=1;RealDigits[N[(Sqrt[75+30*Sqrt[5]]/15)*R,175]]

polrootsreal(45*x^4-30*x^2+1)[4] \\ Charles R Greathouse IV, Sep 02 2024

sqrt(6/sqrt(5)+3)/3 \\ Charles R Greathouse IV, Sep 02 2024

Sqrt(75 + 30*sqrt(5))/15.

Offset corrected, keyword:cons added by R. J. Mathar, Jul 11 2010

A179452 Decimal expansion of sqrt(5 + 2*sqrt(5))/2, the height of a regular pentagon and midradius of an icosidodecahedron with side length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.
Height of a regular pentagon with side length 1. - Jared Kish, Oct 16 2014
Volume of a regular decagonal prism with unit side length and height 2. - Wesley Ivan Hurt, May 04 2021

			1.53884176858762670128514528801845491200335107176889621351957812518743...

Eric Weisstein's World of Mathematics, Icosidodecahedron
Eric Weisstein's World of Mathematics, Pentagon
Index entries for algebraic numbers, degree 4

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451.

Maple
```
sqrt(5+2*sqrt(5.))/2
```

RealDigits[Sqrt[5+2Sqrt[5]]/2,10,120][[1]] (* Harvey P. Dale, Jun 23 2017 *)

PARI
```
sqrt(5+2*sqrt(5))/2
```

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

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

Edited by M. F. Hasler, Oct 16 2014

A179552 Decimal expansion of the volume of pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.

Also equals the covariance-matrix eigenvalue of the regular icosahedron with unit edge lengths. - Chittaranjan Pardeshi, Jul 18 2025

			0.3015028323958245706837155695304698431433590983171469051779540518921...

Wikipedia, Covariance matrix.
Wikipedia, Regular icosahedron.
Wolfram Alpha, Johnson solid 2
Index entries for algebraic numbers, degree 2

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452.

Mathematica
```
RealDigits[N[(5+Sqrt[5])/24,200]]
```

(5+sqrt(5))/24 \\ Charles R Greathouse IV, Oct 30 2023

Equals (5+sqrt(5))/24.

A102769 Decimal expansion of the volume of a dodecahedron with each edge of unit length.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 10 2005

Equals 5*phi^3/(2*xi^2), phi being the golden ratio (A001622) and xi its associate (A182007). - Stanislav Sykora, Nov 23 2013

			7.663118960624631968716053920...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Dodecahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001622 (phi), A182007 (phi associate), A020829 (regular tetrahedron volume), A131594 (regular octahedron volume), A102208 (regular icosahedron volume).

evalf((15+7*sqrt(5))/4,100); # Wesley Ivan Hurt, Jan 29 2017

RealDigits[(Sqrt[5]/2)*(GoldenRatio)^4, 10, 50][[1]] (* G. C. Greubel, Jul 06 2017 *)

(7*sqrt(5)+15)/4 \\ Charles R Greathouse IV, Apr 25 2016

Equals (15 + 7 sqrt(5)) / 4.

Equals (sqrt(5)/2)*(phi)^4, where phi is the golden ratio. - G. C. Greubel, Jul 06 2017

A179449 Decimal expansion of the volume of great icosahedron with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Great icosahedron: 20 faces, 12 vertices, and 30 edges.

			0.31830500937508762649617763802863490189974235016186428155379281441228294...

Eric Weisstein's World of Mathematics, Great Icosahedron
Wikipedia, Great Icosahedron
Index entries for algebraic numbers, degree 2

Cf. A010527, A102208, A179290, A179292, A179294.

RealDigits[N[5*(Sqrt[5]-3)/12, 105]][[1]]

Digits of 5/12 * (3-sqrt(5)).

Partially rewritten by Charles R Greathouse IV, Feb 02 2011

cp's OEIS Frontend

A001622 Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.

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A019881 Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).

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A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.

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A179294 Decimal expansion of radius of inscribed sphere about a regular icosahedron with edge = 1.

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A179587 Decimal expansion of the volume of square cupola with edge length 1.

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A179292 Decimal expansion of radius of inscribed sphere of an icosahedron with radius of circumscribed sphere = 1.

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