A104956 -id:A104956 - cp's OEIS frontend

A256854 Decimal expansion of area of a regular 11-gon with unit edge length.

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

Offset: 1

Stanislav Sykora, Apr 12 2015

			9.36563990694543752488235845328433428788257496183502738768931...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Hendecagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 10

Cf. A000796, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A256853 (p=9), A178816 (p=10), A178809 (p=12).

RealDigits[11/4 Cot[Pi/11],10,120][[1]] (* Harvey P. Dale, Apr 03 2016 *)

p=11; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Equals (p/4)*cot(Pi/p), with p = 11.

A020821 Decimal expansion of 1/8.

Original entry on oeis.org

1, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Also, decimal expansion of Integral_{x=0..oo} x^2/cosh(Pi*x) dx. - Bruno Berselli, Mar 20 2013
Also, decimal expansion of Sum_{i>=1} 1/9^i. - Bruno Berselli, Jan 03 2014
For any triangle ABC, sin(A/2) * sin(B/2) * sin(C/2) <= 1/8, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 16 2022

			0.12500000000000000000...

Kiran S. Kedlaya, A < B, (1999), Problem 6.3, p. 6.
Srinivasa Ramanujan, Question 629, Journal of the Indian Mathematical Society, Vol. 7 (1918), p. 40.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A002194, A104956 (other trigonometric inequalities).

PadRight[{1, 2, 5}, 100] (* Paolo Xausa, Aug 27 2024 *)

Equals Sum_{k>=1} exp(-Pi*k^2) * (Pi*k^2 - 1/4) (Ramanujan, 1918). - Amiram Eldar, Jan 01 2025

A152623 Decimal expansion of 3/2.

Original entry on oeis.org

1, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Oct 30 2009

Sum of the inverses of the tetrahedral numbers (A000292). - Michael B. Porter, Nov 27 2017

For any triangle ABC, cos A + cos B + cos C <= 3/2; equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 17 2022

			1.5000000000000000000000000000000000000000000000000000000000...

Kiran S. Kedlaya, A < B, (1999), Problem 6.2, p. 6.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000292 (tetrahedral numbers).
Sums of inverses: A002117 (cubes), A175577 (octahedral numbers), A295421 (dodecahedral numbers), A175578 (icosahedral numbers).
Cf. A002194, A020821, A104956 (other trigonometric inequalities).

3/2. \\ Charles R Greathouse IV, Jan 10 2022

A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 30 2024

			0.133974596215561353236276829247063816528597373...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Sagitta
Index entries for algebraic numbers, degree 2.

Essentially the same as A334843.
Cf. A010527 (apothem), A104956 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375068 (pentagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A010527, A019913, A152422, A332133.

First[RealDigits[Tan[Pi/12]/2, 10, 100]]

tan(Pi/12)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(4*x^2-8*x+1)[1] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/12)/2 = A019913/2.
Equals 1 - sqrt(3)/2 = 1 - A010527.
Equals A152422^2 = (1 - A332133)^2. - Hugo Pfoertner, Jul 30 2024
Equals A334843-1/2. - R. J. Mathar, Aug 02 2024

A104954 Decimal expansion of the area of the regular triangle with circumradius 1.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005

Equivalently, the area in the complex plane of the smallest convex set containing all order-3 roots of unity.

			1.299038105676657970145584756129404275207103940357785471...

Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
I. J. Zucker and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319.
Index entries for algebraic numbers, degree 2

Cf. A104955, A104956, A104957.

RealDigits[N[Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 3, 108]][[1]]

Equals 3*sqrt(3)/4.
Equals (1/2)*A104956.
Equals 2F1(1/6,1/3 ; 1/2 ; 25/27). [Zucker] - R. J. Mathar, Jun 24 2024

Edited by Georg Fischer, Jul 29 2021

A104955 Decimal expansion of the area of the regular 5-gon (pentagon) of circumradius = 1.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005

Equivalently, the area in the complex plane of the smallest convex set containing all order-5 roots of unity.

			2.377641290737883930291098333448455358514246585314375556118264111075382925212...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
Index entries for algebraic numbers, degree 4.

Cf. A104954, A104956, A104957.

Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 5
RealDigits[(5(Sqrt[(5+Sqrt[5])/2]))/4,10,120][[1]] (* Harvey P. Dale, Jul 21 2013 *)

(5*sqrt((5 + sqrt(5))/2))/4 \\ Michel Marcus, Feb 24 2023

Equals (5*sqrt((5 + sqrt(5))/2))/4.

A104957 Decimal expansion of the area of the regular 7-gon (heptagon) of circumradius = 1.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005

Equivalently, the area in the complex plane of the smallest convex set containing all order-7 roots of unity.

The second largest root of 4096*x^6 - 87808*x^4 + 537824*x^2 - 823543 = 0. [Corrected by Sean A. Irvine, May 24 2025]

			2.736410188638104330479555843359202125813170815480406351432222353...

Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
Index entries for algebraic numbers, degree 6.

Cf. A104954, A104955, A104956.

RealDigits[Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 7, 10, 100][[1]]
RealDigits[Root[-823543 + 537824*#1^2 - 87808*#1^4 + 4096*#1^6 &, 5, 0], 10, 100][[1]]

3*sin((2*Pi)/7) - sin((12*Pi)/7)/2 \\ Michel Marcus, Feb 25 2023

Equals 3*sin(2*Pi/7) - sin(12*Pi/7)/2.
Equals 7*cos(3*Pi/14)/2. - Amiram Eldar, Feb 25 2023
A root of 4096*x^6 -87808*x^4 +537824*x^2 -823543=0. - R. J. Mathar, Aug 29 2025

A178818 Decimal expansion of the diameter of the regular 7-gon (heptagon) of edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

			2.07652139657233656716353886148584033070572020662596852408341737686302...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 6.

Cf. A090488, A102771, A104956, A120011, A178809, A178816, A178817.

SetDefaultRealField(RealField(100)); R:=RealField(); Cot(Pi(R)/7); // G. C. Greubel, Jan 22 2019

evalf[120](cot(Pi/7)); # Muniru A Asiru, Jan 22 2019

Mathematica
```
RealDigits[Cot[Pi/7],10, 100][[1]]
```

default(realprecision, 100); cotan(Pi/7) \\ G. C. Greubel, Jan 22 2019

numerical_approx(cot(pi/7), digits=100) # G. C. Greubel, Jan 22 2019

Digits of cot(Pi/7).

Largest of the 6 real-valued roots of 7*x^6 -35*x^4 +21*x^2 -1=0. - R. J. Mathar, Aug 29 2025

A176325 Decimal expansion of (5+3*sqrt(3))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of (5+3*sqrt(3))/2 is A010721.

a(n) = A104956(n) for n > 2.

			5.09807621135331594029...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002194 (decimal expansion of sqrt(3)), A104956 (decimal expansion of (3*sqrt(3))/2), A010721 (repeat 5, 10).

SetDefaultRealField(RealField(100)); (5+3*Sqrt(3))/2; // G. C. Greubel, Dec 05 2019

evalf( (5+3*sqrt(3))/2, 100); # G. C. Greubel, Dec 05 2019

RealDigits[(5+3Sqrt[3])/2,10,120][[1]] (* Harvey P. Dale, May 20 2011 *)

default(realprecision, 100); (5+3*sqrt(3))/2 \\ G. C. Greubel, Dec 05 2019

numerical_approx((5+3*sqrt(3))/2, digits=100) # G. C. Greubel, Dec 05 2019

A255606 Integer part of the area of a hexagon with side length n.

Original entry on oeis.org

2, 10, 23, 41, 64, 93, 127, 166, 210, 259, 314, 374, 439, 509, 584, 665, 750, 841, 937, 1039, 1145, 1257, 1374, 1496, 1623, 1756, 1893, 2036, 2184, 2338, 2496, 2660, 2829, 3003, 3182, 3367, 3556, 3751, 3951, 4156, 4367, 4583, 4803, 5029, 5261, 5497, 5739, 5985, 6237, 6495

Offset: 1

Kival Ngaokrajang, Feb 27 2015

Column 4 of A255604.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A104956, A255604.

Table[IntegerPart[(6*n^2/(4*Tan[Pi/6]))], {n, 50}] (* Michael De Vlieger, Mar 18 2015 *)
Floor[3/2 Sqrt[3] Range[50]^2] (* Harvey P. Dale, Aug 25 2025 *)

{for(n=1,100,a=floor(6*n^2/(4*tan(Pi/6)));print1(a,", "))}

a(n) = sqrtint(27*n^4)>>1; \\ Kevin Ryde, May 07 2021

a(n) = floor(6*n^2/(4*tan(Pi/6))), n >= 1.

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A256854 Decimal expansion of area of a regular 11-gon with unit edge length.

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A020821 Decimal expansion of 1/8.

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A152623 Decimal expansion of 3/2.

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A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

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A104954 Decimal expansion of the area of the regular triangle with circumradius 1.

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A104955 Decimal expansion of the area of the regular 5-gon (pentagon) of circumradius = 1.

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A104957 Decimal expansion of the area of the regular 7-gon (heptagon) of circumradius = 1.

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