A019918 -id:A019918 - cp's OEIS frontend

A375152 Decimal expansion of the apothem (inradius) of a regular 9-gon with unit side length.

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

Offset: 1

Paolo Xausa, Aug 01 2024

			1.3737387097273111393808320132488363588759362995854...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 6.

Cf. A375151 (circumradius), A375153 (sagitta), A256853 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A019879, A019918, A019968.

Mathematica
```
First[RealDigits[Cot[Pi/9]/2, 10, 100]]
```

 5/tan(Pi/9) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/9)/2 = A019968/2.
Equals 1/(2*tan(Pi/9)) = 1/(2*A019918).
Equals A375151*cos(Pi/9) = A375151*A019879.
Equals A375151 - A375153.
Largest of the 6 real-valued roots of 192*x^6 -432*x^4 +132*x^2 -1=0. - R. J. Mathar, Aug 29 2025

A381154 Decimal expansion of the isoperimetric quotient of a regular 9-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.959050541873609358074543306708643413020181580975...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019918, A256853.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(9*Tan[Pi/9]), 10, 100]]

Equals Pi/(9*tan(Pi/9)) = Pi/(9*A019918).

Equals (4/81)*Pi*A256853.

A019908 Decimal expansion of tangent of 10 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 80 degrees. - Mohammad K. Azarian, Jun 30 2013

			0.176326980708464973471090386868618986121633...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 6.

Cf. A019918.

RealDigits[Tan[10 Degree],10,120][[1]] (* Harvey P. Dale, Oct 14 2012 *)

tan(Pi/18) \\ Hugo Pfoertner, Aug 04 2024

A root of 3*x^6 -27*x^4 +33*x^2 -1 =0 (others A019968, A019948). - R. J. Mathar, Aug 29 2025

tan(Pi/18) = A019819/A019889. - R. J. Mathar, Aug 31 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

A343057 Decimal expansion of tan(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.098491403357164253077197...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8.

Cf. A343055 (sin(Pi/32)), A343056 (cos(Pi/32)).

tan(Pi/m): A002194 (m=3), A019934 (m=5), A020760 (m=6), A343058 (m=7), A188582 (m=8), A019918 (m=9), A019916 (m=10), A019913 (m=12), A343059 (m=14), A019910 (m=15), A343060 (m=16), A343061 (m=17), A019908 (m=18), A019907 (m=20), A343062 (m=24), A019904 (m=30), A343057 (m=32), A019903 (m=36).

R:= RealField(125); Tan(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Tan[Pi/32], 10, 120, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/32)
```

sqrt((2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))))

numerical_approx(tan(pi/32), digits=125) # G. C. Greubel, Sep 30 2022

Equals sqrt( (2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))) ).

cp's OEIS Frontend

A375152 Decimal expansion of the apothem (inradius) of a regular 9-gon with unit side length.

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A381154 Decimal expansion of the isoperimetric quotient of a regular 9-gon.

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A019908 Decimal expansion of tangent of 10 degrees.

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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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A343057 Decimal expansion of tan(Pi/32).

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Formula

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