A004169 -id:A004169 - cp's OEIS frontend

A003401 Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285

Offset: 1

N. J. A. Sloane, R. K. Guy

The terms 1 and 2 correspond to degenerate polygons.
These are also the numbers for which phi(n) is a power of 2: A209229(A000010(a(n))) = 1. - Olivier Gérard Feb 15 1999
From Stanislav Sykora, May 02 2016: (Start)
The sequence can be also defined as follows: (i) 1 is a member. (ii) Double of any member is also a member. (iii) If a member is not divisible by a Fermat prime F_k then its product with F_k is also a member. In particular, the powers of 2 (A000079) are a subset and so are the Fermat primes (A019434), which are the only odd prime members.
The definition is too restrictive (though correct): The Georg Mohr - Lorenzo Mascheroni theorem shows that constructibility using a straightedge and a compass is equivalent to using compass only. Moreover, Jean Victor Poncelet has shown that it is also equivalent to using straightedge and a fixed ('rusty') compass. With the work of Jakob Steiner, this became part of the Poncelet-Steiner theorem establishing the equivalence to using straightedge and a fixed circle (with a known center). A further extension by Francesco Severi replaced the availability of a circle with that of a fixed arc, no matter how small (but still with a known center).
Constructibility implies that when m is a member of this sequence, the edge length 2*sin(Pi/m) of an m-gon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)
If x,y are terms, and gcd(x,y) is a power of 2 then x*y is also a term. - David James Sycamore, Aug 24 2024

			34 is a term of this sequence because a circle can be divided into exactly 34 parts. 7 is not.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 183.
Allan Clark, Elements of Abstract Algebra, Chapter 4, Galois Theory, Dover Publications, NY 1984, page 124.
Duane W. DeTemple, "Carlyle circles and the Lemoine simplicity of polygon constructions." The American Mathematical Monthly 98.2 (1991): 97-108. - N. J. A. Sloane, Aug 05 2021
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. L. van der Waerden, Modern Algebra. Unger, NY, 2nd ed., Vols. 1-2, 1953, Vol. 1, p. 187.

Amiram Eldar, Table of n, a(n) for n = 1..10136 (terms below 10^100; terms 1..2000 from T. D. Noe)
Laura Anderson, Jasbir S. Chahal and Jaap Top, The last chapter of the Disquisitiones of Gauss, arXiv:2110.01355 [math.HO], 2021.
Wayne Bishop, How to construct a regular polygon, Amer. Math. Monthly 85(3) (1978), 186-188.
Alessandro Chiodo, A note on the construction of the Śrī Yantra, Sorbonne Université (Paris, France, 2020).
T. Chomette, Construction des polygones réguliers (in French).
Duane W. DeTemple, Carlyle circles and the Lemoine simplicity of polygon constructions, Amer. Math. Monthly 98(2) (1991), 97-108.
Bruce Director, Measurement and Divisibility.
David Eisenbud and Brady Haran, Heptadecagon and Fermat Primes (the math bit), Numberphile video (2015).
Mauro Fiorentini, Construibili (numeri).
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 463. Original (in Latin).
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag. 63(1) (1990), 3-20. [Annotated scanned copy] [DOI]
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Johann Gustav Hermes, Über die Teilung des Kreises in 65537 gleiche Teile (About the division of the circle into 65537 equal pieces), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Vol. 3 (1894), 170-186.
Friedrich Julius Richelot, De resolutione algebraica aequationis X^257 = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata (On the resolution of the algebraic equation X^257 = 1, or ...), Journal für die reine und angewandte Mathematik 9 (1832), 1-26.
Pierre Wantzel, Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas (Investigations into means of knowing if a problem of geometry can be solved with a straightedge and compass), Journal de Mathématiques Pures et Appliquées 2 (1837), 366-372.
Eric Weisstein's World of Mathematics, Constructible Number.
Eric Weisstein's World of Mathematics, Constructible Polygon.
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Trigonometry.
Eric Weisstein's World of Mathematics, Trigonometry Angles.
Wikipedia, Constructible polygon.
Wikipedia, Johann Gustav Hermes.
Wikipedia, Friedrich Julius Richelot.
Wikipedia, Mohr-Mascheroni theorem.
Wikipedia, Pierre Wantzel.
Wikipedia, Poncelet-Steiner theorem.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.

Subsequence of A295298. - Antti Karttunen, Nov 27 2017
A004729 and A051916 are subsequences. - Reinhard Zumkeller, Mar 20 2010
Cf. A000079, A004169, A000215, A099884, A019434 (Fermat primes).
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17), A272536 (20).
Positions of zeros in A293516 (apart from two initial -1's), and in A336469, positions of ones in A295660 and in A336477 (characteristic function).
Cf. also A046528.
Cf. A000079, A092506.

a003401 n = a003401_list !! (n-1)
a003401_list = map (+ 1) $ elemIndices 1 $ map a209229 a000010_list
-- Reinhard Zumkeller, Jul 31 2012

Select[ Range[ 1300 ], IntegerQ[ Log[ 2, EulerPhi[ # ] ] ]& ] (* Olivier Gérard Feb 15 1999 *)
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Take[ Union[ Flatten[ NestList[2# &, Times @@@ Table[ UnrankSubset[n, Join[{1}, Table[2^2^i + 1, {i, 0, 4}]]], {n, 63}], 11]]], 60] (* Robert G. Wilson v, Jun 11 2005 *)
nn=10; logs=Log[2,{2,3,5,17,257,65537}]; lim2=Floor[nn/logs[[1]]]; Sort[Reap[Do[z={i,j,k,l,m,n}.logs; If[z<=nn, Sow[2^z]], {i,0,lim2}, {j,0,1}, {k,0,1}, {l,0,1}, {m,0,1}, {n,0,1}]][[2,1]]]
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 1300; Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}] (* Robert G. Wilson v, Jul 28 2014 *)

for(n=1,10^4,my(t=eulerphi(n));if(t/2^valuation(t,2)==1,print1(n,", "))); \\ Joerg Arndt, Jul 29 2014

is(n)=n>>=valuation(n,2); if(n<7, return(n>0)); my(k=logint(logint(n,2),2)); if(k>32, my(p=2^2^k+1); if(n%p, return(0)); n/=p; unknown=1; if(n%p==0, return(0)); p=0; if(is(n)==0, 0, "unknown [has large Fermat number in factorization]"), 4294967295%n==0) \\ Charles R Greathouse IV, Jan 09 2022

is(n)=n>>=valuation(n,2); 4294967295%n==0 \\ valid for n <= 2^2^33, conjecturally valid for all n; Charles R Greathouse IV, Jan 09 2022

from sympy import totient
A003401_list = [n for n in range(1,10**4) if format(totient(n),'b').count('1') == 1]
# Chai Wah Wu, Jan 12 2015

Terms from 3 onward are computable as numbers such that cototient-of-totient equals the totient-of-totient: Flatten[Position[Table[co[eu[n]]-eu[eu[n]], {n, 1, 10000}], 0]] eu[m]=EulerPhi[m], co[m]=m-eu[m]. - Labos Elemer, Oct 19 2001, clarified by Antti Karttunen, Nov 27 2017
Any product of 2^k and distinct Fermat primes (primes of the form 2^(2^m)+1). - Sergio Pimentel, Apr 30 2004, edited by Franklin T. Adams-Watters, Jun 16 2006
If the well-known conjecture that there are only five prime Fermat numbers F_k=2^{2^k}+1, k=0,1,2,3,4 is true, then we have exactly: Sum_{n>=1} 1/a(n)= 2*Product_{k=0..4} (1+1/F_k) = 4869735552/1431655765 = 3.40147098978.... - Vladimir Shevelev and T. D. Noe, Dec 01 2010
log a(n) >> sqrt(n); if there are finitely many Fermat primes, then log a(n) ~ k log n for some k. - Charles R Greathouse IV, Oct 23 2015

Definition clarified by Bill Gosper. - N. J. A. Sloane, Jun 14 2020

A255241 Decimal expansion of 2cos(3Pi/7).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 13 2015

This is also the decimal expansion of 2*sin(Pi/14).
rho_2 := 2*cos(3*Pi/7) and rho(7) := 2*cos(Pi/7) (see A160389) are the two positive zeros of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(7), the length ratio of the smaller diagonal and the side in the regular 7-gon (heptagon). See A187360 and a link to the arXiv paper given there, eq. (20) for the zeros of C(n, x). The other zero is negative, rho_3 := 2*cos(5*Pi/n). See -A255249.
Also the edge length of a regular 14-gon with unit circumradius. Such an m-gon is not constructible using a compass and a straightedge (see A004169). With an even m, in fact, it would be constructible only if the (m/2)-gon were constructible, which is not true in this case (see A272487). - Stanislav Sykora, May 01 2016

			0.445041867912628808577805128993589518932711137529089910623974031...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 3

Cf. A004169, A160389, A187360, A255249, A232736, A255240.

Edge lengths of other nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A130880 (n=18), A272491 (n=19). - Stanislav Sykora, May 01 2016

R:= RealField(120); 2*Cos(3*Pi(R)/7); // G. C. Greubel, Sep 04 2022

RealDigits[N[2Cos[3Pi/7], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/14)
```

polrootsreal(x^3 - x^2 - 2*x + 1)[2] \\ Charles R Greathouse IV, Oct 30 2023

numerical_approx(2*cos(3*pi/7), digits=120) # G. C. Greubel, Sep 04 2022

2*cos(3*Pi/7) = 2*sin(Pi/14) = 2*A232736 = 1/A231187 = 0.4450418679...
See A232736 for the decimal expansion of cos(3*Pi/7) = sin(Pi/14).
Equals i^(6/7) - i^(8/7). - Peter Luschny, Apr 04 2020
From Peter Bala, Oct 11 2021: (Start)
Equals 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)), where z = exp(2*Pi*i/7).
Equals 1 - A255240. (End)

Offset corrected by Stanislav Sykora, May 01 2016

A004729 Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass).

Original entry on oeis.org

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

Offset: 0

N. J. A. Sloane

The 32 divisors of the product of the 5 known Fermat primes.
The only known odd numbers whose totient is a power of 2. - Labos Elemer, Dec 06 2000
Equals first 32 members of A001317. Also, equals first 32 members of A053576. - Omar E. Pol, Dec 10 2008
Omitting the first term a(0)=1 gives A045544 (the number of sides of constructible odd-sided regular polygons.)

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 140.

Eric Weisstein's World of Mathematics, Regular Polygon, Sierpiński Sieve, Constructible Polygon
OEIS Wiki, Constructible odd-sided polygons
OEIS Wiki, Sierpinski's triangle
Index entries for sequences related to divisors of numbers

Essentially same as A045544.

Cf. A000010, A000215, A001317, A003401, A003527, A004169, A004729, A019434, A045544, A047999, A053576, A054432.

Mathematica
```
Divisors[2^32-1]
```
PARI
```
divisors(1<<32-1)
```

Edited by Daniel Forgues, Jun 17 2011

A130880 Decimal expansion of 2*sin(Pi/18).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 26 2007

Also: a bond percolation threshold probability on the triangular lattice.

Also: the edge length of a regular 18-gon with unit circumradius. Such an m-gon is not constructible using a compass and a straightedge (see A004169). With an even m, in fact, it would be constructible only if the (m/2)-gon were constructible, which is not true in this case (see A272488). - Stanislav Sykora, May 01 2016

			0.347296355333860697703433253538629592...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.18.1, p. 373.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Steven R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat., Vol. 3, Issue 2-4 (1999), pp. 323-335.
Wikipedia, Constructible number.
Wikipedia, Percolation threshold.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 3.

Cf. A004169, A019819, A019829, A019889, A063289, A133749, A178959, A332437, A332438.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A272491 (n=19). - Stanislav Sykora, May 01 2016

RealDigits[N[2Sin[Pi/18], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/18)
```

Equals 2*A019819 = A019829/A019889.
Algebraic number with minimal polynomial over Q equal to x^3 - 3*x + 1, a cyclic cubic, having zeros 2*sin(Pi/18) (= 2*cos(4*Pi/9)), 2*sin(5*Pi/18) (= 2*cos(2*Pi/9)) and -2*sin(7*Pi/18) (= -2*cos(Pi/9)). Cf. A332437. - Peter Bala, Oct 23 2021
Equals 2 + rho(9) - rho(9)^2, an element of the extension field Q(rho(9)), with rho(9) = 2*cos(Pi/9) = A332437 with minimal polynomial x^3 - 3*x - 1 over Q. - Wolfdieter Lang, Sep 20 2022
Equals -1 + Product_{k>=3} (1 - (-1)^k/A063289(k)). - Amiram Eldar, Nov 22 2024
Equals A133749/2 = 1 - A178959. - Hugo Pfoertner, Dec 15 2024

A086592 Denominators in left-hand half of Kepler's tree of fractions.

Original entry on oeis.org

2, 3, 3, 4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 8, 8, 6, 6, 9, 9, 10, 10, 11, 11, 9, 9, 12, 12, 11, 11, 13, 13, 7, 7, 11, 11, 13, 13, 14, 14, 13, 13, 17, 17, 15, 15, 18, 18, 11, 11, 16, 16, 17, 17, 19, 19, 14, 14, 19, 19, 18, 18, 21, 21, 8, 8, 13, 13, 16, 16, 17, 17, 17, 17, 22, 22, 19, 19, 23

Offset: 1

Antti Karttunen, Aug 28 2003

Form a tree of fractions by beginning with 1/1 and then giving every node i/j two descendants labeled i/(i+j) and j/(i+j).
Level n of the left-hand half of the tree consists of 2^(n-1) nodes: 1/2; 1/3, 2/3; 1/4, 3/4, 2/5, 3/5; 1/5, 4/5, 3/7, 4/7, 2/7, 5/7, 3/8, 5/8; ... .
The right-hand half is identical to the left-hand half. - Michel Dekking, Oct 05 2017
n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (Franklin T. Adams-Watters' comment), that is the sequence obtained by adding numerator and denominator in the Calkin-Wilf enumeration system of positive rationals. A020650(n)/A020651(n) is also an enumeration system of all positive rationals (Yu-Ting system), and in each level m >= 0 (ranks between 2^m and 2^(m+1)-1) rationals are the same in both systems. Thus a(n) has the same terms in each level as A007306. The same property occurs in all numerator+denominator sequences of enumeration systems of positive rationals, as, for example, A007306 (A007305+A047679), A071585 (A229742+A071766), and A268087 (A162909+A162910). - Yosu Yurramendi, Apr 06 2016

Johannes Kepler, Mysterium cosmographicum, Tuebingen, 1596, 1621, Caput XII.
Johannes Kepler, Harmonice Mundi, Linz, 1619, Liber III, Caput II.
Johannes Kepler, The Harmony of the World [1619], trans. E. J. Aiton, A. M. Duncan and J. V. Field, American Philosophical Society, Philadelphia, 1997, p. 163.

Johannes Kepler, Harmonices mundi libri V ... (A Latin original scanned in Internet Archive. The fraction-tree is illustrated on the page 27 of the third book (Liber III), which is on the page 117 of the PDF-document.)
Johannes Kepler, Excerpt from the Chapter II of the Book III of the Harmony of the World: On the seven harmonic divisions of the string (Illustrates the A020651/A086592-tree).
OEIS Wiki, Historical sequences
Pelegrí Viader, Jaume Paradís and Lluís Bibiloni, A New Light on Minkowski's ?(x) Function, J. Number Theory, 73 (2) (1998), 212-227. See p. 215.
Index entries for fraction trees
Index entries for sequences related to music

Bisection of A020650.
See A093873/A093875 for the full tree.
A020651 gives the numerators. Bisection: A086593. Cf. A002487, A004169.

(* b = A020650 *) b[1] = 1; b[2] = 2; b[3] = 1; b[n_] := b[n] = Switch[ Mod[n, 4], 0, b[n/2 + 1] + b[n/2], 1, b[(n - 1)/2 + 1], 2, b[(n - 2)/2 + 1] + b[(n - 2)/2], 3, b[(n - 3)/2]]; a[n_] := b[2n]; Array[a, 100] (* Jean-François Alcover, Jan 22 2016 *)

maxlevel <- 15
d <- c(1,2)
for(m in 0:maxlevel)
for(k in 1:2^m) {
   d[2^(m+1)    +k] <- d[k] + d[2^m+k]
   d[2^(m+1)+2^m+k] <- d[2^(m+1)+k]
}
b <- vector()
for(m in 0:maxlevel) for(k in 0:(2^m-1)) b[2^m+k] <- d[2^(m+1)+k]
a <- vector()
for(n in 1:2^maxlevel) {a[2*n-1] <- b[n]; a[2*n] <- b[n+1]}
a[1:128]
# Yosu Yurramendi, May 16 2018

a(n) = A020650(n) + A020651(n) = A020650(2n).
a(n) = A071585(A059893(n)), a(A059893(n)) = A071585(n), n > 0. - Yosu Yurramendi, May 30 2017
a(2*n-1) = A086593(n); a(2*n) = A086593(n+1), n > 0. - Yosu Yurramendi, May 16 2018
a(n) = A007306(A231551(n)), n > 0. - Yosu Yurramendi, Aug 07 2021

Entry revised by N. J. A. Sloane, May 24 2004

A272487 Decimal expansion of the edge length of a regular heptagon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 01 2016

The edge length e(m) of a regular m-gon is e(m) = 2*sin(Pi/m). In this case, m = 7, and the constant, a = e(7), is the smallest m for which e(m) is not constructible using a compass and a straightedge (see A004169). With an odd m, in fact, e(m) would be constructible only if m were a Fermat prime (A019434).

			0.8677674782351162409515366656967175092199814555749197528890946...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Heptagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A004169, A019434.
Cf. A160389.
Edge lengths of nonconstructible n-gons: A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

N[2*Sin[Pi/7], 25] (* G. C. Greubel, May 01 2016 *)
RealDigits[2*Sin[Pi/7],10,120][[1]] (* Harvey P. Dale, Mar 07 2020 *)

PARI
```
2*sin(Pi/7)
```

Equals 2*sin(Pi/7) = 2*cos(Pi*5/14).
Equals i^(-5/7) + i^(5/7). - Gary W. Adamson, Feb 12 2022
One of the 6 real-valued roots of x^6 -7*x^4 +14*x^2 -7 =0. - R. J. Mathar, Aug 29 2025

A272488 Decimal expansion of the edge length of a regular 9-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 01 2016

The edge length e(m) of a regular m-gon is e(m) = 2*sin(Pi/m). In this case, m = 9, and the constant, a = e(9), is not constructible using a compass and a straightedge (see A004169). With an odd m, in fact, e(m) would be constructible only if m were a Fermat prime (A019434).

			0.6840402866513374660881992293645191615261667350283212569300969953...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A004169, A019434.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

RealDigits[N[2Sin[Pi/9], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/9)
```

Equals 2*sin(Pi/9) = 2*cos(Pi*7/18) = 2*A019829.
Equals Im((4+4*sqrt(3)*i)^(1/3)). - Gerry Martens, Mar 19 2024
A root of x^6 -6*x^4 +9*x^2 -3 =0. - R. J. Mathar, Aug 29 2025

cp's OEIS Frontend

A003401 Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.

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A255241 Decimal expansion of 2*cos(3*Pi/7).

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A004729 Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass).

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A130880 Decimal expansion of 2*sin(Pi/18).

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A086592 Denominators in left-hand half of Kepler's tree of fractions.

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A272487 Decimal expansion of the edge length of a regular heptagon with unit circumradius.

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A255241 Decimal expansion of 2cos(3Pi/7).