A090488 -id:A090488 - cp's OEIS frontend

A156035 Decimal expansion of 3 + 2*sqrt(2).

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

Offset: 1

Klaus Brockhaus, Feb 02 2009

Limit_{n -> oo} b(n+1)/b(n) = 3+2*sqrt(2) for b = A155464, A155465, A155466.
Limit_{n -> oo} b(n)/b(n-1) = 3+2*sqrt(2) for b = A001652, A001653, A002315, A156156, A156157, A156158. - Klaus Brockhaus, Sep 23 2009
From Richard R. Forberg, Aug 14 2013: (Start)
Ratios b(n+1)/b(n) for all sequences of the form b(n) = 6*b(n-1) - b(n-2), for any initial values of b(0) and b(1), converge to this ratio.
Ratios b(n+1)/b(n) for all sequences of the form b(n) = 5*b(n-1) + 5*b(n-2) + b(n-3), for all b(0), b(1) and b(2) also converge to 3 + 2*sqrt(2). For example see A084158 (Pell Triangles).
Ratios of alternating values, b(n+2)/b(n), for all sequences of the form b(n) = 2*b(n-1) + b(n-2), also converge to 3 + 2*sqrt(2). These include A000129 (Pell Numbers). Also see A014176. (End)
Let ABCD be a square inscribed in a circle. When P is the midpoint of the arc AB, then the ratio (PC*PD)/(PA*PB) is equal to 3+2*sqrt(2). See the Mathematical Reflections link. - Michel Marcus, Jan 10 2017
Limit of ratios of successive terms of A001652 when n-> infinity. - Harvey P. Dale, Jun 16 2017; improved by Bernard Schott, Feb 28 2022
A quadratic integer with minimal polynomial x^2 - 6x + 1. - Charles R Greathouse IV, Jul 11 2020
Ratio between radii of the large circumscribed circle R and the small internal circle r drawn on the Sangaku tablet at Isaniwa Jinjya shrine in Ehime Prefecture (pictures in links). - Bernard Schott, Feb 25 2022

			3 + 2*sqrt(2) = 5.828427124746190097603377448...

Diogo Queiros-Condé and Michel Feidt, Fractal and Trans-scale Nature of Entropy, Iste Press and Elsevier, 2018, page 45.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Mathematical Reflections, Solution to Problem J286, Issue 1, 2014, p. 5.
Bernard Schott, Sangaku at Isaniwa Jinya, The six circles.
Terakoya Suzu, Sangaku (mathematics tablet) II, Sangaku at Isaniwa Jinya shrine.
Wikipedia, Sangaku.
Bernard Ycart, Les Sangakus, Sangaku du Temple Isaniwa Jinya (in French).
Index entries for algebraic numbers, degree

Cf. A002193 (sqrt(2)), A090488, A010466, A014176.
Cf. A155464, A155465, A155466.
Cf. A104178 (decimal expansion of log_10(3+2*sqrt(2))).
Cf. A000129, A001109, A001541, A001542, A001652, A001653, A002315, A005319, A075870, A038723, A038725, A038761, A054488, A054489, A075848, A077413, A084158, A106328, A156156, A156157, A156158.
Cf. A242412 (sangaku).

SetDefaultRealField(RealField(100));  3 + 2*Sqrt(2); // G. C. Greubel, Aug 21 2018

RealDigits[N[3+2*Sqrt[2],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

3+sqrt(8) \\ Charles R Greathouse IV, Jun 10 2011

Equals 1 + A090488 = 3 + A010466. - R. J. Mathar, Feb 19 2009
Equals exp(arccosh(3)), since arccosh(x) = log(x+sqrt(x^2-1)). - Stanislav Sykora, Nov 01 2013
Equals (1+sqrt(2))^2, that is, A014176^2. - Michel Marcus, May 08 2016
The periodic continued fraction is [5; [1, 4]]. - Stefano Spezia, Mar 17 2024

A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

Original entry on oeis.org

2, 5, 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

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

Equivalently, the area in the complex plane of the smallest convex set containing all order-6 roots of unity.
Subtracting 2.5 (i.e., dropping the first two digits) we obtain 0.09807.... which is a limiting mean cluster density for a bond percolation model at probability 1/2 [Finch]. - R. J. Mathar, Jul 26 2007
This constant is also the minimum radius of curvature of the exponential curve (occurring at x = -log(2)/2 = -0.34657359...). - Jean-François Alcover, Dec 19 2016
Luminet proves that this is the critical impact parameter of a bare black hole, in multiples of the Schwarzschild radius. That is, light from a distant source coming toward a black hole is captured by the black hole at smaller distances and deflected at larger distances. - Charles R Greathouse IV, May 21 2022
For any triangle ABC, sin(A) + sin(B) + sin(C) <= 3*sqrt(3)/2, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 16 2022
Surface area of a triangular bipyramid (Johnson solid J_12) with unit edges. - Paolo Xausa, Aug 04 2025

			2.59807621135331594029116951225880855041420788071557094208371046917789952536320...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24, p. 412.

G. C. Greubel, Table of n, a(n) for n = 1..10000
S. R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.
Kiran S. Kedlaya, A < B, (1999), Problem 6.1, p. 6.
J.-P. Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics, vol. 75, no. 1-2 (May 1979), pp. 228-235.
Michael Penn, not as bad as it seems..., YouTube video, 2021.
Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
Eric Weisstein's World of Mathematics, Twenty-Vertex Entropy Constant.
Wikipedia, Hexagon.
Wikipedia, Regular polygon.
Wikipedia, Triangular bipyramid.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A104954, A104955, A104957.

Cf. Areas of other regular polygons: A120011, A102771, A178817, A090488, A256853, A178816, A256854, A178809.

Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 6
RealDigits[(3*Sqrt[3])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

3*sqrt(3)/2 \\ G. C. Greubel, Jul 03 2017

Equals (3*sqrt(3))/2, that is, 2*A104954.

Equals Product_{k>=3} (((k - 1)^2*(k + 2))/((k + 1)^2*(k - 2)))^(k/2). - Antonio Graciá Llorente, Oct 13 2024

A120011 Decimal expansion of sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Eric Desbiaux, Jul 04 2008

Area of equilateral triangle of side 1.
Quadratic number with denominator 4 and minimal polynomial 16x^2 - 3. - Charles R Greathouse IV, Jun 30 2021
With offset 1, surface area of a pentagonal bipyramid (Johnson solid J_13) with unit edges. - Paolo Xausa, Aug 04 2025

			0.43301270189221932338186158537646809173570131345259515701395....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Pentagonal bipyramid.
Index entries for algebraic numbers, degree 2

Cf. A010527.

Cf. Areas of higher regular polygons: A102771, A104956, A178817, A090488, A256853, A178816, A256854, A178809.

RealDigits[Sqrt[3]/4,10,120][[1]] (* Harvey P. Dale, Feb 18 2016 *)

sqrt(3)/4 \\ Charles R Greathouse IV, Jun 26 2013

A090388 Decimal expansion of 1 + sqrt(3).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

1 + sqrt(3) is the length of the minimal Steiner network that connects the four vertices of a unit square. - Lekraj Beedassy, May 02 2008
This is the case n = 12 in the identity (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1 + 2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012
Equals n + n/(n + n/(n + n/(n + ...))) for n = 2. - Stanislav Sykora, Jan 23 2014
A non-optimal solution to the problem of finding the length of shortest fence that protects privacy of a square garden [Kawohl]. Cf. A256965. - N. J. A. Sloane, Apr 14 2015
Perimeter of a 30-60-90 triangle with longest leg equal to 1. - Wesley Ivan Hurt, Apr 09 2016
Length of the second shortest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Dec 13 2020
Surface area of a square pyramid (Johnson solid J_1) with unit edges. - Paolo Xausa, Aug 04 2025

			2.7320508075688772...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Bernd Kawohl, Some nonconvex shape optimization problems, in: Optimal Shape Design, Eds. A.Cellina u. A. Ornelas, Springer Lecture Notes in Math.1740 (2000), p. 7-46.
Ian Stewart, Pursuing Polygonal Privacy, Mathematical Recreations Column, Scientific American, 284 (No. 2, 2001), 88-89.
Wikipedia, Square pyramid.
Index entries for algebraic numbers, degree 2

Cf. n + n/(n + n/(n + ...)): A090458 (n = 3), A090488 (n = 4), A090550 (n = 5), A092294 (n = 6), A092290 (n = 7), A090654 (n = 8), A090655 (n = 9), A090656 (n = 10). - Stanislav Sykora, Jan 23 2014

Cf., also A256965.

Digits:=100: evalf((1+sqrt(3))); # Wesley Ivan Hurt, Apr 09 2016

RealDigits[1 + Sqrt[3], 10, 100][[1]] (* Alonso del Arte, Feb 23 2014 *)

1 + sqrt(3) \\ Michel Marcus, Apr 10 2016

Equals 1 + A002194. - R. J. Mathar, Oct 16 2015

Equals A019973 -1 . - R. J. Mathar, May 25 2023

Better definition from Rick L. Shepherd, Jul 02 2004

A102771 Decimal expansion of area of a regular pentagon with unit edge length.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 10 2005

			1.720477400588966922759011977...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagon
Index entries for algebraic numbers, degree 4

Cf. Areas of other regular polygons: A120011, A104956, A178817, A090488, A256853, A178816, A256854, A178809.

RealDigits[(5/4)*Sqrt[GoldenRatio^3/Sqrt[5]], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

5/(4*tan(Pi/5)) \\ Michel Marcus, Mar 25 2015

Equals sqrt(25 + 10*sqrt(5)) / 4.
Equals (3*phi+1)*sqrt(3-phi) with the golden section phi = (1 + sqrt(5))/2. - Wolfdieter Lang, Jan 25 2013
Equals 5/(4*tan(Pi/5)). - Michel Marcus, Mar 25 2015
Equals (5/4)*sqrt(phi^3/sqrt(5)). - G. C. Greubel, Jul 03 2017

Corrected the title. - Stanislav Sykora, Apr 12 2015

A090550 Decimal expansion of solution to n/x = x - n for n = 5.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

n/x = x - n with n = 1 gives the Golden Ratio = 1.6180339887...

Equals n + n/(n + n/(n + n/(n + ....))) for n = 5. See also A090388. - Stanislav Sykora, Jan 23 2014

			5.85410196624968454...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Index entries for algebraic numbers, degree 2.

Cf. n + n/(n + n/(n + ...)): A090388 (n = 2), A090458 (n = 3), A090488 (n = 4), A092294 (n = 6), A092290 (n = 7), A090654 (n = 8), A090655 (n = 9), A090656 (n = 10). - Stanislav Sykora, Jan 23 2014

Cf. A001622, A057088.

RealDigits[(5 + 3 Sqrt[5])/2, 10, 120][[1]] (* Harvey P. Dale, Nov 27 2013 *)

(5 + 3*sqrt(5))/2 \\ G. C. Greubel, Jul 03 2017

n/x = x - n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 5: x = (5 + sqrt(45))/2 = 5.85410196624968454...
Equals (5 + 3*sqrt(5))/2 = 1 + 3*phi = sqrt(5)*(phi)^2, where phi is the golden ratio. - G. C. Greubel, Jul 03 2017
Equals 2*phi^3 - phi^2. - Michel Marcus, Apr 20 2020
Minimal polynomial is x^2 - 5x - 5 (this number is an algebraic integer). - Alonso del Arte, Apr 20 2020(n).
Equals lim_{n->oo} A057088(n+1)/A057088(n) = 1 + 3*phi. - Wolfdieter Lang, Nov 16 2023

A178809 Decimal expansion of the area of the regular 12-gon (dodecagon) of edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

Surface area of a regular hexagonal prism with unit side length and height. - Wesley Ivan Hurt, May 04 2021

			11.196152422706631880582339024517617100828415761431141884167420938355...

G. C. Greubel, Table of n, a(n) for n = 2..10000
Index entries for algebraic numbers, degree 2

Cf. A010482, A090488, A104956, A120011, A176532.

Mathematica
```
RealDigits[N[6+3*Sqrt[3],150]][[1]]
```

sqrt(27)+6 \\ Charles R Greathouse IV, Apr 25 2016

Equals 6+3*sqrt(3).

Equals 1 + A176532 = 6 + A010482. - R. J. Mathar, Jun 25 2010

Offset corrected and keyword:cons inserted by R. J. Mathar, Jun 25 2010

A090458 Decimal expansion of (3 + sqrt(21))/2.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Decimal expansion of the solution to n/x = x-n for n-3. n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...
n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 3: x = (3 + sqrt(21))/2 = 3.79128784747792...
x=3.7912878474... is the shape of a rectangle whose geometric partition (as at A188635) consists of 3 squares, then 1 square, then 3 squares, etc., matching the continued fraction of x, which is [3,1,3,1,3,1,3,1,3,1,...]. (See the Mathematica program below.) - Clark Kimberling, May 05 2011
x appears to be the limit for n to infinity of the ratio of the number of even numbers that take n steps to reach 1 to the number of odd numbers that take n steps to reach 1 in the Collatz iteration. As A005186(n-1) is the number of even numbers that take n steps to reach 1, this means x = lim A005186(n-1)/A176866(n). - Markus Sigg, Oct 20 2020
From Wolfdieter Lang, Sep 02 2022: (Start)
This integer in the quadratic number field Q(sqrt(21)) equals the (real) cube root of 27 + 6*sqrt(21) = 54.4954541... . See Euler, Elements of Algebra, Article 748 or Algebra (in German) p. 306, Kapitel 12, 187.
Subtracting 3 from the present number gives the (real) cube root of
  -27 +  6*sqrt(21) = 0.4954541... . (End)

			3.79128784747792...

Leonhard Euler, Vollständige Anleitung zur Algebra, (1770), Reclam, Leipzig, 1883, p.306, Kapitel 12, 187.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Leonhard Euler, Elements of Algebra, p. 244, Article 748.
Markus Sigg, Heuristic argument for the asymptotic value of the even/odd ratio of number of steps in the Collatz iteration, 2020.
Index entries for algebraic numbers, degree 2

Of the same type as this: A090388 (n=2), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10).
Equals 3*A176014 (constant).
Cf. A356034.

FromContinuedFraction[{3,1,{3,1}}]
ContinuedFraction[%,20]
RealDigits[N[%%,120]] (*A090458*)
N[%%%,40]
RealDigits[(3 + Sqrt[21])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

solve(x=3,4,x^2-3*x-3) \\ Charles R Greathouse IV, Oct 04 2011

(3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011

Equals (27 + 6*sqrt(21))^(1/3). - Wolfdieter Lang, Sep 01 2022

Additional comments from Rick L. Shepherd, Jul 02 2004

A178817 Decimal expansion of the area of the regular 7-gon (heptagon) of edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

			3.63391244400158899253619300760022057873501036159544491714598040951029...

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

Cf. A178818, A343058.

Cf. Areas of other regular polygons: A120011, A102771, A104956, A090488, A256853, A178816, A256854, A178809.

SetDefaultRealField(RealField(100)); R:=RealField(); 7*Cot(Pi(R)/7)/4; // G. C. Greubel, Jan 22 2019

evalf[120]((7/4)*cot(Pi/7)); # Muniru A Asiru, Jan 22 2019

Mathematica
```
RealDigits[7*Cot[Pi/7]/4, 10, 100][[1]]
```

p=7; a=(p/4)*cotan(Pi/p)  \\ Set realprecision in excess. - Stanislav Sykora, Apr 12 2015

numerical_approx(7*cot(pi/7)/4, digits=100) # G. C. Greubel, Jan 22 2019

Equals (7/4) * cot(Pi/7).
From Michal Paulovic, Dec 27 2022: (Start)
Equals 7 / (4 * tan(Pi/7)) = 7 / (4 * A343058).
Equals sqrt(7/3 * (35 + 2 * 196^(1/3) * ((13 - 3 * sqrt(3) * i)^(1/3) + (13 + 3 * sqrt(3) * i)^(1/3)))) / 4.
Equals sqrt(7/4) * sqrt(35/12 + (637/54 - sqrt(-2401/108))^(1/3) + (637/54 + sqrt(-2401/108))^(1/3)).
(End)
A root of the polynomial 4096*x^6 - 62720*x^4 + 115248*x^2 - 16807. - Joerg Arndt, Jan 02 2023

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

cp's OEIS Frontend

A156035 Decimal expansion of 3 + 2*sqrt(2).

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A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

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A120011 Decimal expansion of sqrt(3)/4.

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A090388 Decimal expansion of 1 + sqrt(3).

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A102771 Decimal expansion of area of a regular pentagon with unit edge length.

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A090550 Decimal expansion of solution to n/x = x - n for n = 5.

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A178809 Decimal expansion of the area of the regular 12-gon (dodecagon) of edge length 1.

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