A130880 -id:A130880 - cp's OEIS frontend

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

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

A004169 Values of m for which a regular polygon with m sides cannot be constructed with ruler and compass.

Original entry on oeis.org

7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91

Offset: 1

N. J. A. Sloane, Branislav Kisacanin (bkisacan(AT)eecs.uic.edu)

Numbers m for which phi(a(m)) is not a power of 2, phi = A000010, Euler's totient function. - Reinhard Zumkeller, Jul 31 2012

Numbers m for which A295660(m) > 1. - Lorenzo Sauras Altuzarra, Nov 04 2018

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 183.
B. L. van der Waerden, Modern Algebra. Unger, NY, 2nd ed., Vols. 1-2, 1953, Vol. 1, p. 187.

T. D. Noe, Table of n, a(n) for n = 1..1000
Claudi Alsina and Roger B. Nelson, A Panoply of Polygons, Dolciani Math. Expeditions Vol. 58, AMS/MAA (2023), see page 16.
C. F. Gauss, Disquisitiones Arithmeticae, Lipsiae, 1801. Reprinted in C. F. Gauss, Werke, 1863.
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 460.

Cf. A003401 (complement).
Cf. A000010, A209229.
Edge lengths of nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

a004169 n = a004169_list !! (n-1)
a004169_list = map (+ 1) $ elemIndices 0 $ map a209229 a000010_list
-- Reinhard Zumkeller, Jul 31 2012

Select[ Range[75], !IntegerQ[ Log[2, EulerPhi[#] ] ]& ] (* Jean-François Alcover, Nov 24 2011, after A003401 *)

is(n)=my(t=4294967295); n>>=valuation(n,2); n/=gcd(n,t); if(gcd(n,t)>1, return(1)); if(n==1, return(0)); if(n<9e2585827972, return(1)); forprime(p=7,1e5, if(n%p==0, return(1))); warning("Result is conjectural on the nonexistence of Fermat primes >= F(33)."); 1 \\ Charles R Greathouse IV, Oct 23 2015

a(n) = n + O(log^2 n). - Charles R Greathouse IV, Oct 23 2015

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

A332438 Decimal expansion of (2*cos(Pi/9))^2 = A332437^2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Mar 31 2020

This algebraic number rho(9)^2 of degree 3 is a root of its minimal polynomial x^3 - 6*x^2 + 9*x - 1.
The other two roots are x2 = (2*cos(5*Pi/9))^2 = (2*cos(4*Pi/9))^2 = (R(4,rho(9)))^2 = 2 - rho(9) = 0.120614758..., and x3 = (2*cos(7*Pi/9))^2 = (2*cos(7*Pi/9))^2 = (R(7,rho(9)))^2 = 4 + rho(9) - rho(9)^2 = 2.347296355... = A130880 + 2, with rho(9) = 2*cos(Pi/9) = A332437, the monic Chebyshev polynomials R (see A127672), and the computation is done modulo the minimal polynomial of rho(9) which is x^3 - 3*x - 1 (see A187360).
This gives the representation of these roots in the power basis of the simple field extension Q(rho(9)). See the linked W. Lang paper in A187360, sect. 4.
This number rho(9)^2 appears as limit of the quotient of consecutive numbers af various sequences, e.g., A094256 and A094829.
The algebraic number rho(9)^2 - 2 = 1.532088898... of Q(rho(9)) has minimal polynomial x^3 - 3*x + 1 over Q. The other roots are -rho(9) = -A332437 and 2 + rho(9) - rho(9)^2 = A130880. - Wolfdieter Lang, Sep 20 2022

			3.5320888862379560704047853011108333478716649...

Index entries for algebraic numbers, degree 3.

Cf. A019879, A063289, A094256, A094829, A127672, A130880, A187360, A332437.

2 + 2*cos(2*Pi/n): A104457 (n = 5), A116425 (n = 7), A296184 (n = 10), A019973 (n = 12).

RealDigits[(2*Cos[Pi/9])^2, 10, 100][[1]] (* Amiram Eldar, Mar 31 2020 *)

(2*cos(Pi/9))^2 \\ Michel Marcus, Sep 23 2022

Equals (2*cos(Pi/9))^2 = rho(9)^2 = A332437^2.
Equals 2 + i^(4/9) - i^(14/9). - Peter Luschny, Apr 04 2020
Equals 2 + w1^(1/3) + w2^(1/3), where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1. - Wolfdieter Lang, Sep 20 2022
Constant c = 2 + 2*cos(2*Pi/9). The linear fractional transformation z -> c - c/z has order 9, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(z))))))))). - Peter Bala, May 09 2024
From Amiram Eldar, Nov 22 2024: (Start)
Equals 3 + sec(Pi/9)/2 = 3 + 1/(2*A019879).
Equals 3 + Product_{k>=3} (1 + (-1)^k/A063289(k)). (End)

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

A215885 a(n) = 3*a(n-1) - a(n-3), with a(0) = 3, a(1) = 3, and a(2) = 9.

Original entry on oeis.org

3, 3, 9, 24, 69, 198, 570, 1641, 4725, 13605, 39174, 112797, 324786, 935184, 2692755, 7753479, 22325253, 64283004, 185095533, 532961346, 1534601034, 4418707569, 12723161361, 36634883049, 105485941578, 303734663373, 874569107070, 2518221379632, 7250929475523

Offset: 0

Roman Witula, Aug 25 2012

The Berndt-type sequence number 5a for the argument 2Pi/9 defined by the first relation from the section "Formula". We see that a(n) is equal to the sum of the n-th negative powers of the c(j) := 2*cos(2*Pi*j/9), j=1,2,4 (the A215664(n) is equal to the respective n-th positive powers, further both sequences can be obtained from the two-sided recurrence relation: X(n+3) = 3*X(n+1) - X(n), n in Z, with X(-1) = X(0) = 3, and X(1) = 0).
From the last formula in Witula's comments to A215664 it follows that 2*(-1)^n*a(n) = A215664(n)^2 - A215664(2*n).
The following decomposition holds true: (X - c(1)^(-n))*(X - c(2)^(-n))*(X - c(4)^(-n)) = X^3 - a(n)*X^2 - (-1)^n*A215664(n)*X - (-1)^n.
For n >= 1, a(n) is the number of cyclic (0,1,2)-compositions of n that avoid the pattern 110 provided the positions of the parts of the composition on the circle are fixed. (Similar comments hold for the pattern 012 and for the pattern 001.) - Petros Hadjicostas, Sep 13 2017
See the Maple program by Edlin and Zeilberger for counting the q-ary cyclic compositions of n that avoid one or more patterns provided the positions of the parts of the composition are fixed on the circle. The program is located at D. Zeilberger's personal website (see links). For the sequence here, q=3 and the pattern is A=110. - Petros Hadjicostas, Sep 13 2017

			For n=3, we have a(3) = 3^3 - 3 = 24 ternary cyclic compositions of n=3 (with fixed positions on the circle for the parts) that avoid 110 because we have to exclude 110, 101, and 011. - _Petros Hadjicostas_, Sep 13 2017

Michael De Vlieger, Table of n, a(n) for n = 0..1999
A. E. Edlin and D. Zeilberger, The Goulden-Jackson cluster method for cyclic words, Adv. Appl. Math. 25 (2000), 228-232.
A. E. Edlin and D. Zeilberger, Maple program; Local copy
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012), 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Cf. A130880, A147704, A215664, A215665, A215666, A274018.

Mathematica
```
LinearRecurrence[{3,0,-1}, {3,3,9}, 50]
```

my(x='x+O('x^30)); Vec(3*(1-2*x)/(1-3*x+x^3)) \\ Altug Alkan, Sep 13 2017

a(n) = 3*A147704(n).
a(n) = c(1)^(-n) + c(2)^(-n) + c(4)^(-n) = (-c(1)*c(2))^n + (-c(1)*c(4))^n + (-c(2)*c(4))^n, where c(j) := 2*cos(2*Pi*j/9).
G.f.: Sum_{n>=0} a(n)*x^n = 3-3*x*(x^2-1)/(1-3*x+x^3) = 3*(1-2*x)/(1-3*x+x^3).
G.f. of Edlin and Zeilberger (2000): 1+Sum_{n>=1} a(n)*x^n = 1-3*x*(x^2-1)/(1-3*x+x^3) = (1-2*x^3)/(1-3*x+x^3). - Petros Hadjicostas, Sep 13 2017
a(n) = ceiling(r^n) for n >= 1, where r = 1/A130880 is the largest root of x^3 - 3*x^2 + 1. - Tamas Lengyel, Feb 20 2022

A272489 Decimal expansion of the edge length of a regular 11-gon with unit circumradius.

Original entry on oeis.org

5, 6, 3, 4, 6, 5, 1, 1, 3, 6, 8, 2, 8, 5, 9, 3, 9, 5, 4, 2, 2, 8, 3, 5, 8, 3, 0, 6, 9, 3, 2, 3, 3, 7, 9, 8, 0, 7, 1, 5, 5, 5, 7, 9, 7, 9, 4, 6, 5, 3, 3, 7, 4, 3, 6, 6, 2, 1, 6, 0, 6, 1, 2, 1, 7, 5, 6, 9, 7, 5, 9, 7, 0, 3, 8, 0, 5, 8, 3, 3, 6, 2, 4, 6, 9, 3, 5, 2, 3, 6, 9, 0, 3, 7, 7, 3, 0, 9, 9, 9, 3, 5, 9, 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 = 11, and the constant, a = e(11), 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).

			0.5634651136828593954228358306932337980715557979465337436621606121...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 10

Cf. A004169, A019434.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

RealDigits[N[2Sin[Pi/11], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/11)
```

Equals 2*sin(Pi/11) = 2*cos(Pi*9/22).

A272491 Decimal expansion of the edge length of a regular 19-gon with unit circumradius.

Original entry on oeis.org

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

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 = 19, and the constant, a = e(19), 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).

			0.32918918056146778828730411817587683902434496671930824670294254...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 18

Cf. A004169, A019434.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18).

RealDigits[N[2Sin[Pi/19], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/19)
```

Equals 2*sin(Pi/19) = 2*cos(Pi*17/38).

A272490 Decimal expansion of the edge length of a regular 13-gon with unit circumradius.

Original entry on oeis.org

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

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 = 13, and the constant, a = e(13), 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).

			0.47863132857511553429750745252042379040634604547661202671031943...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 12

Cf. A004169, A019434.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A255241 (n=14), A130880 (n=18), A272491 (n=19).

RealDigits[N[2Sin[Pi/13], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/13)
```

Equals 2*sin(Pi/13) = 2*cos(Pi*11/26).

A147704 Diagonal sums of Riordan array ((1-2x)/(1 - 3x + x^2),x(1-x)/(1 - 3x + x^2)).

Original entry on oeis.org

1, 1, 3, 8, 23, 66, 190, 547, 1575, 4535, 13058, 37599, 108262, 311728, 897585, 2584493, 7441751, 21427668, 61698511, 177653782, 511533678, 1472902523, 4241053787, 12211627683, 35161980526, 101244887791, 291523035690, 839407126544, 2416976491841, 6959406439833

Offset: 0

Paul Barry, Nov 10 2008

Diagonal sums of A147703.
Hankel transform is := 1,2,3,0,0,0,0,0,0,0,... - Philippe Deléham, Dec 15 2008
For n -> infinity, a(n+1)/a(n) -> 2.87938... = 1/A130880 = the largest diagonal of a nonagon (9-gon) with side 1 (see Redondo & Huylebrouck); compare to the F(n+1)/F(n) -> 1.618... = A001622 = the golden section or diagonal of a pentagon with side 1, where F is the Fibonacci sequence A000045. - Dirk Huylebrouck, Feb 15 2015

A. Redondo Buitrago and D. Huylebrouck, Nonagons in the Hagia Sophia and the Selimiye Mosque, Nexus Network Journal on Architecture and Mathematics, January 2015.
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Cf. A000045, A001622, A130880.

LinearRecurrence[{3,0,-1},{1,1,3},30] (* Harvey P. Dale, May 24 2016 *)

Vec((1-x^2)/(1-3*x+x^3) + O(x^20)) \\ Michel Marcus, Feb 16 2015

G.f.: (1-x^2)/(1 - 3x + x^3).
a(n) = 3*a(n-1) - a(n-3), n>2 ; a(0)=1, a(1)=1, a(2)=3. - Philippe Deléham, Dec 15 2008
a(n) = (floor(A^n)+1)/3 for n>=1 where A = 2.8793... is the largest root of x^3-3x^2+1. - Stephen Bartell, Aug 15 2024

cp's OEIS Frontend

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

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A004169 Values of m for which a regular polygon with m sides cannot be constructed with ruler and compass.

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

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A332438 Decimal expansion of (2*cos(Pi/9))^2 = A332437^2.

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

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A215885 a(n) = 3*a(n-1) - a(n-3), with a(0) = 3, a(1) = 3, and a(2) = 9.

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A272489 Decimal expansion of the edge length of a regular 11-gon with unit circumradius.

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