A019676 -id:A019676 - cp's OEIS frontend

A073010 Decimal expansion of Pi/sqrt(27).

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

Original name: Decimal expansion of Sum_{n>0} 1/(n*binomial(2*n,n)).
This appears to be Pi/sqrt(27). See A111510. - Marco Matosic, Feb 27 2008
This is Pi*sqrt(3)/9 = A019676*A002194, see eq. (12) in Lehmer link. - R. J. Mathar, Mar 04 2009
Value of the Dirichlet L-series of the non-principal character modulo m=3 (A102283) at s=1. - R. J. Mathar, Oct 03 2011
Construct the largest possible circle inside a given equilateral triangle. This constant is the ratio of the area of the circle to the area of the triangle (A245670 is analogous square in triangle). - Rick L. Shepherd, Jul 29 2014

			0.60459978807807261686469275254738524409468...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (81), page 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005), 25-36.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova, Computation of the Kummer ratio of the class number for prime cyclotomic fields, arXiv:1908.01152 [math.NT], 2019. See r(q) for q=3 in Table 1, p. 7.
D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449-457. See eq. (12).
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991), 53-55.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (1.6.2)
Simon Plouffe, Sum(1/(n*binomial(2*n,n)), n=1..infinity), see p. 87.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 501.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for transcendental numbers.

Cf. A002194, A019676, A111510, A245670, A248682.

R:=RealField(106); SetDefaultRealField(R); n:=Pi(R)/Sqrt(27); Reverse(Intseq(Floor(10^105*n))); // Bruno Berselli, Mar 12 2018

RealDigits[ N [Sum[1/(n*Binomial[2n, n]), {n, 1, Infinity}], 110]] [[1]]
RealDigits[Pi/(3*Sqrt[3]), 10, 105][[1]] (* T. D. Noe, Sep 11 2013 *)

Pi/sqrt(27) \\ Charles R Greathouse IV, Sep 11 2013

-Pi/(3*sqrt(3)) = Sum_{n>=0} (1/(6*n+1) - 2/(6*n+2) - 3/(6*n+3) - 1/(6*n+4) + 2/(6*n+5) + 3/(6*n+6)). - Mats Granvik, Sep 08 2013
Equals Integral_{0..oo} 2*x/((x^2+1)*(x^4+x^2+1)) dx. - Jean-François Alcover, Sep 10 2013
From Peter Bala, Feb 16 2015: (Start)
Pi/sqrt(27) = Sum_{n >= 0} 1/((3*n + 1)*(3*n + 2)) = 1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ....
Continued fraction: 1/(1 + 1^2/(1 + 2^2/(2 + 4^2/(1 + 5^2/(2 + ... + (3*n + 1)^2/(1 + (3*n + 2)^2/(2 + ... ))))))).
Pi/sqrt(27) = Integral_{t = 0..1/2} 1/(t^2 - t + 1) dt = Integral_{t = 0..1/2} (1 + t - t^3 - t^4)/(1 - t^6) dt.
Pi/sqrt(27) = (1/4)*Sum_{n >= 0} (-1/8)^n * (9*n + 5)/((3*n + 1)*(3*n + 2)).
BBP-type formulas:
Pi/sqrt(27) = Sum_{n >= 0} (1/64)^(n+1)*( 32/(6*n + 1) + 16/(6*n + 2) - 4/(6*n + 4) - 2/(6*n + 5) ) follows from the above integral representation.
Pi/sqrt(27) = Sum_{n >= 0} (-1)^n*(1/27)^(n+1)*( 9/(6*n + 1) + 9/(6*n + 2) + 6/(6*n + 3) + 3/(6*n + 4) + 1/(6*n + 5) ) follows from the result: Pi/3 = Integral_{t = 0..1/sqrt(3)} 1/(1 - sqrt(3)*t + t^2) dt. (End)
Equals Integral_{x=0..oo} x*I_0(x)*K_0(x)^2 dx over a triple product of modified Bessel functions. - R. J. Mathar, Oct 14 2015
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(exp(x) + exp(-x) + 1) dx.
Equals Integral_{x=0..oo} 1/(1 + x + x^2 + x^3 + x^4 + x^5) dx. (End)
Equals (3*S - 4)/8, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Product_{p prime} (1 - Kronecker(-3, p)/p)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p)^(-1). - Amiram Eldar, Nov 06 2023
From Peter Bala, Dec 09 2023: (Start)
Pi/sqrt(27) = Sum_{n >= 1} 1/(n*binomial(2*n,n)) = (1/6)*Sum_{n >= 1} 3^n/(n*binomial(2*n,n)) (see Lehmer, equation 12, and also p. 456).
Pi/sqrt(27) = (1/2)*Sum_{n >= 0} 1/((2*n + 1)*binomial(2*n,n)).
Pi/sqrt(27) = (9/4)*Sum_{n >= 3} (n - 1)*(n - 2)/binomial(2*n,n). (End)
Equals integral_{x=0..oo} 1/(1-x^3) dx [Nahin]. - R. J. Mathar, May 16 2024
From Peter Bala, Mar 05 2025: (Start)
Equals 2*Integral_{x = 0..1} 1/(3 + x^2) dx = Integral_{x = 0..1} (4 - x)/(sqrt(x)*(12 + x*(1 - x))) dx.
Equals Sum_{n >= 1} (-1/3)^n * (3 - 14*n)/(n*(2*n-1)*binomial(4*n, 2*n)). The series terms are O(7*sqrt(2*Pi/n)/48^n). (End)
Equals Integral_{x=0..oo} (x^3)/(x^6 + 1) dx. - Kritsada Moomuang, Jun 04 2025

A056020 Numbers that are congruent to +-1 mod 9.

Original entry on oeis.org

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset: 1

Robert G. Wilson v, Jun 08 2000

Or, numbers k such that k^2 == 1 (mod 9).

Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

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

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).
Cf. A129805 (primes), A195042 (partial sums).
Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887, A332437.
Cf. A381319 (general case mod n^2).

a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]
(* or *)
LinearRecurrence[{1, 1, -1}, {1, 8, 10}, 67] (* Mike Sheppard, Feb 18 2025 *)

a(n)=9*(n>>1)+if(n%2,1,-1) \\ Charles R Greathouse IV, Jun 29 2011

for(n=1,40,print1(9*n-8,", ",9*n-1,", ")) \\ Charles R Greathouse IV, Jun 29 2011

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).
a(n+1) - a(n) = A010697(n). (End)
a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = a(n-2) + 9, for n > 2.
a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/9) (A332437).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/9)*cosec(Pi/9). (End)
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) ~ (3^2/2)*n. (End)

A256853 Decimal expansion of the area of a unit 9-gon.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 12 2015

From Michal Paulovic, May 09 2024: (Start)
This constant multiplied by the square of the side length of a regular enneagon equals the area of that enneagon.
9^2 divided by this constant equals 36 * tan(Pi/9) = 13.10292843... which is the perimeter and the area of an equable enneagon with its side length 4 * tan(Pi/9) = 1.45588093... . (End)

			6.181824193772900127213744059619763614941713348134358098386864...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Nonagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A000796, A019669, A019670, A019673, A019676, A019685, A019968, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A178816 (p=10), A256854 (p=11), A178809 (p=12).

evalf(9 / (4 * tan(Pi/9)), 100); # Michal Paulovic, May 09 2024

RealDigits[(9/4)*Cot[Pi/9], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

p=9; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Equals (p/4)*cot(Pi/p), with p = 9.
From Michal Paulovic, May 09 2024: (Start)
Equals 9 * sqrt(2 / (1 - sin(5 * A000796 / 18)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019669 / 9)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019670 / 6)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019673 / 3)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019676 / 2)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(50 * A019685)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * Pi / 18)) - 1) / 4.
Equals 9 * sqrt(4 / (2 - i^(4/9) - i^(-4/9)) - 1) / 4.
Equals 9 * sqrt(1 / (8 - (-32 + sqrt(-3072))^(1/3) - (-32 - sqrt(-3072))^(1/3)) - 1/16). (End)
Largest of the 6 real-valued roots of 4096*x^6 -186624*x^4 +1154736*x^2 -177147 =0. - R. J. Mathar, Aug 29 2025

A353410 a(n) = (tan(1Pi/9))^(2n) + (tan(2Pi/9))^(2n) + (tan(3Pi/9))^(2n) + (tan(4Pi/9))^(2n).

Original entry on oeis.org

4, 36, 1044, 33300, 1070244, 34420356, 1107069876, 35607151476, 1145248326468, 36835122753252, 1184744167077204, 38105444942929620, 1225602095970073572, 39419576386043222340, 1267869080483029127412, 40779027899804602385460, 1311593714249667915837060, 42185362424185765127267748

Offset: 0

Bernard Schott, Apr 17 2022

Sum_{k=1..(m-1)/2} (tan(k*Pi/m))^(2*n) is an integer when m >= 3 is an odd integer (see AMM link); this sequence is for the case m = 9.

Note tan(3*Pi/9) = tan(Pi/3) = sqrt(3).

			a(1) = tan^2 (Pi/9) + tan^2 (2*Pi/9) + tan^2 (3*Pi/9) + tan^2 (4*Pi/9) = 36.

Michel Bataille and Li Zhou, A Combinatorial Sum Goes on Tangent, The American Mathematical Monthly, Vol. 112, No. 7 (Aug. - Sep., 2005), Problem 11044, pp. 657-659.
Index entries for linear recurrences with constant coefficients, signature (36,-126,84,-9).

Similar with: A000244 (m=3), 2*A165225 (m=5), A108716 (m=7), this sequence (m=9), A275546 (m=11), A353411 (m=13).
Cf. A019676 (Pi/9), A019918 (tan(Pi/9)), A019938 (tan(2*Pi/9)).
Cf. A215948.

LinearRecurrence[{36, -126, 84, -9}, {4, 36, 1044, 33300}, 18] (* Amiram Eldar, Apr 18 2022 *)

G.f.: 4*(1 - 27x + 63*x^2 - 21*x^3)/((1 - 3*x)*(1 - 33*x + 27*x^2 - 3*x^3)). - Stefano Spezia, Apr 18 2022

a(n) = A215948(n) + 3^n. - Jianing Song, Apr 19 2022

More terms from Stefano Spezia, Apr 18 2022

A019696 Decimal expansion of 2*Pi/9.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The area of a deltoid formed by rolling a circle with radius 1/3 inside a unit-radius circle. - Amiram Eldar, Nov 16 2021

Volume shared by a sphere of radius 1 inscribed in a cube and one of the six pyramids inscribed in the cube. - Omar E. Pol, Sep 01 2024

			0.69813170079773183076947630739544508537714875541669...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Deltoid.
Index entries for transcendental numbers

Equals twice A019676.

RealDigits[(2*Pi)/9,10,120][[1]] (* Harvey P. Dale, Aug 24 2013 *)

PARI
```
2*Pi/9 \\ Michel Marcus, Nov 16 2021
```

A019705 Decimal expansion of sqrt(Pi)/3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers.

Cf. A002161, A019676.

RealDigits[Sqrt[Pi]/3,10,120][[1]] (* Harvey P. Dale, Jun 07 2018 *)

sqrt(Pi)/3 \\ Charles R Greathouse IV, Oct 01 2022

Equals sqrt(A019676). - Michel Marcus, Aug 31 2014

Equals 10 * Sum_{k>=0} (k+1/2)!/(k+4)!. - Amiram Eldar, Jun 19 2023

cp's OEIS Frontend

A073010 Decimal expansion of Pi/sqrt(27).

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A056020 Numbers that are congruent to +-1 mod 9.

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A256853 Decimal expansion of the area of a unit 9-gon.

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A353410 a(n) = (tan(1*Pi/9))^(2*n) + (tan(2*Pi/9))^(2*n) + (tan(3*Pi/9))^(2*n) + (tan(4*Pi/9))^(2*n).

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A019696 Decimal expansion of 2*Pi/9.

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A019705 Decimal expansion of sqrt(Pi)/3.

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A353410 a(n) = (tan(1Pi/9))^(2n) + (tan(2Pi/9))^(2n) + (tan(3Pi/9))^(2n) + (tan(4Pi/9))^(2n).