A010527 -id:A010527 - cp's OEIS frontend

A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

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

Offset: 1

Paolo Xausa, Jul 26 2024

			1.0382606982861682835817694307429201653528601033...

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. A374957 (circumradius), A374972 (sagitta), A178817 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A178818, A073052, A343058.

Mathematica
```
First[RealDigits[Cot[Pi/7]/2, 10, 100]]
```

 5/(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/7)/2 = A178818/2.
Equals 1/(2*tan(Pi/7)) = 1/(2*A343058).
Equals A374957*cos(Pi/7) = A374957*A073052.
Equals A374957 - A374972.
Largest of the 6 real-valued roots of 448*x^6 -560*x^4 +84*x^2 -1 =0. - R. J. Mathar, Aug 29 2025

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

Original entry on oeis.org

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

A375191 Decimal expansion of the apothem (inradius) of a regular 11-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 04 2024

			1.702843619444625004524065173324424415978649993...

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 10.

Cf. A375190 (circumradius), A375192 (sagitta), A256854 (area).

Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375193 (12-gon).

First[RealDigits[Cot[Pi/11]/2, 10, 100]]

 5/tan(Pi/11) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/11)/2.
Equals 1/(2*tan(Pi/11)).
Equals A375190*cos(Pi/11).
Equals A375190 - A375192.

A375193 Decimal expansion of the apothem (inradius) of a regular 12-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 04 2024

Apart from the first digit the same as A010527.

			1.8660254037844386467637231707529361834714026269...

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 2.

Cf. A188887 (circumradius), A375194 (sagitta), A178809 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon).
Cf. A019884, A019913, A019973, A332133, A375069.

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

sqrt(3)/2 + 1 \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/12)/2 = (2 + sqrt(3))/2 = A019973/2.
Equals 1/(2*tan(Pi/12)) = 1/(2*A019913).
Equals A188887*cos(Pi/12) = A188887*A019884.
Equals A188887 - A375194.
Equals A332133^2 = 2 - A375069. - Hugo Pfoertner, Aug 04 2024

A084765 a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 49, 4801, 46099201, 4250272665676801, 36129635465198759610694779187201, 2610701117696295981568349760414651575095962187244375364404428801

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jun 04 2003

Product_{k>=1} (1 + 1/a(k)) = sqrt(3/2) (see A010527).
A subsequence of A001079 (cf. formula), which must contain any prime occurring in A001079. The initial term a(0)=1 seems rather unnatural; using the recurrence relation it would yield the constant sequence 1,1,1,... Note that this sequence corresponds to sequence b(n) in Shallit's paper, which starts only at offset n=1. - M. F. Hasler, Sep 27 2009
Since if x is even (x^2-2)/2 = 2*y^2-1 and 10 is even from a(1) onward this is a reduced version of the LL sequence starting with 10 (A135927) as it is reduced by dividing by 2 it is also the difference between two possible LL sequences. - Roderick MacPhee, May 31 2015
For n >= 3, a(n) == 201 (mod 1000) if n is even, a(n) == 801 (mod 1000) if n is odd. - Robert Israel, Jun 01 2015
The next term -- a(8) -- has 128 digits. - Harvey P. Dale, Mar 28 2020

G. C. Greubel, Table of n, a(n) for n = 0..10
Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
H. S. Wilf, Limit of a sequence, Elementary Problem E 1093, Amer. Math. Monthly 61 (1954), 424-425.

Cf. A001079, A002812, A010527, A084764, A135927.

[n le 2 select 5^(n-1) else 2*Self(n-1)^2-1: n in [1..10]]; // Vincenzo Librandi, Jun 02 2015

1,seq(expand((5+2*sqrt(6))^(2^n)+(5-2*sqrt(6))^(2^n))/2, n=0..10); # Robert Israel, Jun 01 2015

a[n_]:= a[n]= If[n<2, 5^n, 2 a[n-1]^2 -1]; Table[a[n], {n,0,10}]
Join[{1}, NestList[2 #^2 - 1 &, 5, 10]] (* Harvey P. Dale, Mar 28 2020 *)

first(m)={my(v=[1,5]);for(i=3,m,v=concat(v, 2*v[i-1]^2 - 1));v;} \\ Anders Hellström, Aug 22 2015

def A084765(n): return 1 if n==0 else chebyshev_T(2^(n-1), 5)
[A084765(n) for n in range(11)] # G. C. Greubel, May 17 2023

a(n+1) = (x^(2^n) + y^(2^n))/2, with x = 5 + 2*sqrt(6), y = 5 - 2*sqrt(6).
a(n) = A001079(2^(n-1)) with a(0) = 1. - M. F. Hasler, Sep 27 2009
4*sqrt(6)/11 = Product_{n >= 1} (1 - 1/(2*a(n))). See A002812 for some general properties of the recurrence a(n+1) = 2*a(n)^2 - 1. - Peter Bala, Nov 11 2012
a(n) = cos(2^(n-1)*arccos(5)) for n >= 1. - Peter Luschny, Oct 12 2022

A171970 Integer part of the height of an equilateral triangle with side length n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 58, 59, 60, 61, 62, 63

Offset: 1

Reinhard Zumkeller, Jan 20 2010

Eric Weisstein's World of Mathematics, Equilateral Triangle
Wikipedia, Equilateral triangle

Beatty sequence of A010527.

Cf. A022838, A004526, A005843, A171971, A171972.

a(n)=sqrtint(3*n^2\4) \\ Charles R Greathouse IV, Jan 06 2013

a(n) = floor(n*sqrt(3)/2).
a(n) = floor(A022838(n)/2).
a(n)*A004526(n) <= A171971(n)
a(n)*A005843(n) <= A171972(n).

A179593 Decimal expansion of the volume of pentagonal rotunda with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

			6.91776296812470206991299603070264133354087600944966144271710443099823...

Wolfram Alpha, Johnson solid 6
Index entries for algebraic numbers, degree 2.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591, A179592.

Mathematica
```
RealDigits[N[(45+17*Sqrt[5])/12,200]]
```

Digits of (45+17*sqrt(5))/12.

A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232738.

			0.9807852804032304491261822361342390369739337308933360950029160885453...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 8

Cf. A232738 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232735 (real(I^(1/7))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

RealDigits[Cos[Pi/16], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

real(I^(1/8)) \\ Seiichi Manyama, Apr 04 2021

cos(Pi/16) \\ Seiichi Manyama, Apr 04 2021

sqrt(2+sqrt(2+sqrt(2)))/2 \\ Seiichi Manyama, Apr 04 2021

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
Root of 128*x^8 -256*x^6 +160*x^4 -32*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
2*this^2 -1 = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8;1/2;1/2). - R. J. Mathar, Aug 31 2025

A019872 Decimal expansion of sine of 63 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

RealDigits[Sin[63 Degree],10,120][[1]] (* Harvey P. Dale, Oct 31 2018 *)

sin(31/60*Pi) \\ Charles R Greathouse IV, Nov 06 2017

Equals A019851 * A019878 + A019830 * A019857 = A010527 * A019896 + A019812 * (1/2). - R. J. Mathar, Jan 27 2021
This^2 + A019836^2=1. - R. J. Mathar, Aug 31 2025
One of the 8 real-valued roots of 256*x^8-512*x^6+304*x^4-48*x^2+1=0. (Other A019890, A019836, A019818) - R. J. Mathar, Aug 31 2025

A019874 Decimal expansion of sine of 65 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 12 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 12

RealDigits[Sin[65 Degree],10,120][[1]] (* Harvey P. Dale, Oct 29 2020 *)

sin(13/36*Pi) \\ Charles R Greathouse IV, Nov 06 2017

Equals cos(5*Pi/36) = 2F1(17/24,7/24;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008

Equals A019861 * A019886 + A019822 * A019847 = A010527 * A019894 + A019814*(1/2). - R. J. Mathar, Jan 27 2021

cp's OEIS Frontend

A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

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

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A375191 Decimal expansion of the apothem (inradius) of a regular 11-gon with unit side length.

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A375193 Decimal expansion of the apothem (inradius) of a regular 12-gon with unit side length.

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A084765 a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=5.

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A171970 Integer part of the height of an equilateral triangle with side length n.

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A179593 Decimal expansion of the volume of pentagonal rotunda with edge length 1.

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