A019913 -id:A019913 - cp's OEIS frontend

A019973 Decimal expansion of tangent of 75 degrees.

3, 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

Offset: 1

N. J. A. Sloane

An equivalent definition of this sequence: decimal expansion of x > 1 satisfying x^2 - 4*x + 1 = 0. - Arkadiusz Wesolowski, Nov 28 2011
An algebraic integer of degree 2 with minimal polynomial x^2 - 4*x + 1. - Charles R Greathouse IV, Oct 17 2016
Length of the second longest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Dec 13 2020

			3.732050807568877293527446341505872366942805253810380628...

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Ivan Panchenko)
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A019824, A019884, A019913.

Cf. A001075, A001353, A001835, A049310.

SetDefaultRealField(RealField(100)); 2 + Sqrt(3); // G. C. Greubel, Nov 20 2018

RealDigits[Tan[75 Degree],10,120][[1]] (* Harvey P. Dale, Nov 08 2011 *)
RealDigits[2+Sqrt[3], 10, 100][[1]] (* G. C. Greubel, Nov 20 2018 *)

sqrt(3)+2 \\ Charles R Greathouse IV, Oct 17 2016

numerical_approx(2+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

Equals 2 + sqrt(3) = 2+A002194 = cotangent of 15 degrees. - Rick L. Shepherd, Jul 04 2004
Equals exp(arccosh(2)). - Amiram Eldar, Aug 07 2023
c^n = A001835(n) + (1 + sqrt(3)) * A001353(n) = A001075(n) + sqrt(3) * A001353(n); where c = 2 + sqrt(3). - Gary W. Adamson, Oct 14 2023
Equals lim_{n->oo} S(n, 4)/ S(n-1, 4), with the S-Chebyshev polynomial (see A049310) S(n, 4) = A001353(n+1). See the A001353 formula from Oct 06 2002 by Gregory V. Richardson. - Wolfdieter Lang, Nov 15 2023
Equals A019884 / A019824. - R. J. Mathar, Jan 12 2024
Equals 1/A019913. - Hugo Pfoertner, Mar 24 2024

Checked by Neven Juric (neven.juric(AT)apis-it.hr), Feb 04 2008

A101263 Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.

Original entry on oeis.org

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

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 25 2005

sqrt(2 - sqrt(3)) is the shape of the lesser sqrt(6)-contraction rectangle, as defined at A188739. - Clark Kimberling, Apr 16 2011
This is a constructible number, since 12-gon is a constructible polygon. See A003401 for more details. - Stanislav Sykora, May 02 2016
It is also smaller positive coordinate of (symmetrical) intersection points of x^2 + y^2 = 4 circle and y = 1/x hyperbola. The bigger coordinate is A188887. - Leszek Lezniak, Sep 18 2018
The greatest possible minimum distance between 8 points in a unit square (Schaer and Meir, 1965; Schaer, 1965; Croft et al., 1991). - Amiram Eldar, Feb 24 2025

			0.517638090205041524697797675248096656698137802639861027628006414630113....

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

G. C. Greubel, Table of n, a(n) for n = 0..5000
J. Schaer and A. Meir, On a geometric extremum problem, Canadian Mathematical Bulletin, Vol. 8, No. 1 (1965), pp. 21-27.
J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
Eric Weisstein's World of Mathematics, Dodecagon.
Index entries for algebraic numbers, degree 4.

Cf. A003401, A019824, A019913, A091999, A120683, A188739, A188887, A329247.

r = 6^(1/2); t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]  (*A101263*)
RealDigits[Sqrt[2-Sqrt[3]],10,120][[1]] (* Harvey P. Dale, Apr 24 2018 *)

2*sin(Pi/12) \\ Stanislav Sykora, May 02 2016

Equals sqrt(A019913). - R. J. Mathar, Apr 20 2009
Equals 2*sin(Pi/12) = 2*cos(Pi*5/12). - Stanislav Sykora, May 02 2016
Equals i^(5/6) + i^(-5/6). - Gary W. Adamson, Jul 07 2022
From Amiram Eldar, Nov 24 2024: (Start)
Equals A120683 / 2 = 2 * A019824 = 1 / A188887 = exp(-A329247).
Equals (sqrt(3)-1)/sqrt(2).
Equals Product_{k>=1} (1 + (-1)^k/A091999(k)). (End)

A092735 Decimal expansion of Pi^7.

Original entry on oeis.org

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

Offset: 4

Mohammad K. Azarian, Apr 12 2004

Wentworth (1903) shows how to compute the tangent of 15 degrees (A019913) to five decimal places by the laborious process of adding up the first few terms of Pi/12 + Pi^3/5184 + 2Pi^5/3732480 + 17Pi^7/11287019520 + ... - Alonso del Arte, Mar 13 2015

			3020.293227776792067514206493...

George Albert Wentworth, New Plane and Spherical Trigonometry, Surveying, and Navigation. Boston: The Atheneum Press (1903): 240.

G. C. Greubel, Table of n, a(n) for n = 4..10000
Index entries for transcendental numbers

Cf. A000796, A002161, A019692, A091925, A092731, A000364, A002437.

R:= RealField(100); (Pi(R))^7; // G. C. Greubel, Mar 09 2018

RealDigits[Pi^7, 10, 100][[1]] (* Alonso del Arte, Mar 13 2015 *)

PARI
```
Pi^7 \\ G. C. Greubel, Mar 09 2018
```

From Peter Bala, Oct 30 2019: (Start)
Pi^7 = (6!/(2*33367)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^7 + 1/(n + 5/6)^7 ), where 33367 = ((3^7 + 1)/4)*A000364(3) = A002437(3).
Pi^7 = (6!/(2*1191391)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^7 - 1/(n + 3/10)^7 - 1/(n + 7/10)^7 + 1/(n + 9/10)^7 ), where 1191391 = ((5^7 - 1)/4)*A000364(3). (End)

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

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

A381157 Decimal expansion of the isoperimetric quotient of a regular 12-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.97704861665685333572562679495712274710387812858570...

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

Cf. A019913, A178809.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon).

First[RealDigits[Pi/(12*Tan[Pi/12]), 10, 100]]

Equals Pi/(12*tan(Pi/12)) = Pi/(12*A019913).

Equals (1/36)*Pi*A178809.

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)))) ).

A379338 Decimal expansion of 2*(2 - sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 21 2024

The greatest possible minimum distance between 7 points in a unit square (Schaer and Meir, 1965; Schaer, 1965; Croft et al., 1991). - Amiram Eldar, Feb 24 2025

			0.535898384862245412945107316988255266114389492379...

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

J. Schaer and A. Meir, On a geometric extremum problem, Canadian Mathematical Bulletin, Vol. 8, No. 1 (1965), pp. 21-27.
J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.

Cf. A002194, A019913.

Mathematica
```
RealDigits[2(2-Sqrt[3]),10,100][[1]]
```

Minimal polynomial: x^2 - 8*x + 4. - Stefano Spezia, Aug 03 2025

A354128 Decimal expansion of 7 - 4*sqrt(3).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, May 18 2022

The smallest root of x^2 - 14*x + 1 = 0.

			0.07179676972449082589021463397651...

Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022, p. 13.

Cf. A002194, A010502, A019913 (square root), A354129 (multiplicative inverse).

Mathematica
```
First[RealDigits[N[7-4Sqrt[3],92]]]
```

Equals (2 - sqrt(3))^2 = A019913^2. - Jianing Song, May 27 2022

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A019973 Decimal expansion of tangent of 75 degrees.

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A101263 Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.

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A092735 Decimal expansion of Pi^7.

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A375069 Decimal expansion of the sagitta of a regular hexagon 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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A381157 Decimal expansion of the isoperimetric quotient of a regular 12-gon.

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A343057 Decimal expansion of tan(Pi/32).

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