A019827 -id:A019827 - cp's OEIS frontend

A019830 Decimal expansion of sine of 21 degrees.

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

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals

Cf. A019812, A019816, A019823, A019827, A019881, A019892, A019885, A019896.

RealDigits[Sin[21 Degree],10,120][[1]] (* Harvey P. Dale, Apr 06 2016 *)

sin(7*Pi/60) \\ Charles R Greathouse IV, Jan 30 2016

Equals A019823*A019892 + A019816*A019885 = A019827*A019896 + A019812*A019881. - R. J. Mathar, Jan 27 2021

A019907 Decimal expansion of tangent of 9 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 81 degrees. - Mohammad K. Azarian, Jun 30 2013

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

RealDigits[Tan[9 Degree],10,120][[1]] (* Harvey P. Dale, Aug 23 2020 *)

Equals tan(Pi/20) = A019818/A019890. - R. J. Mathar, Aug 29 2025
Smallest positive of the 8 real-valued roots of x^8-44*x^6+166*x^4-44*x^2+1=0. (Others A019925, A019961, A019979). - R. J. Mathar, Aug 31 2025
Equals A019827/(1+A019881). - R. J. Mathar, Sep 06 2025

A340725 Decimal expansion of Gamma(9/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.06862870211931935489..

DLMF, Gamma function properties, DLMF section 5.5.
Index to sequences related to gamma function.

Cf. A019827, A246745, A256191, A340722.

Maple
```
evalf(GAMMA(9/10),120) ;
```

RealDigits[Gamma[9/10], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A256191 = Pi/A019827 . [DLMF (5.5.3)]

A246745 * this *2^(3/10) /sqrt(2*Pi) = A340722 . [DLMF (5.5.5)]

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

			0.09801714032956060199419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
imag(I^(1/16))
```
PARI
```
sin(Pi/32)
```
PARI
```
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).
One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025
Equals A232738/(2*A343056). - R. J. Mathar, Sep 05 2025

A267860 An infinite ternary 3-Fibonacci sequence (replace each 00 factor of the Fibonacci word with 020).

Original entry on oeis.org

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

Offset: 0

Mahdi Saleh, Apr 07 2016

nonn

A word constructed by replacing each 00 factor of the Fibonacci word (A003849) with 020. The obtained ternary sequence is a word with Sturmian erasures (by removing each word,the obtained binary sequence is Sturmian)[1]. By removing each of 0's or 2's, the set of replacements on the Fibonacci word, is equal to the morphisms of deriving the Fibonacci word [2]. So the obtained binary word by removing each of 0's,1's or 2's is the Fibonacci word. Since the slope of the sequential projection (sending for example one letter to 1 and all the others to 0) is 1, the factor complexity of this ternary word for each integer n>0, is n+2.[3]
The binary sequence obtained by removing all 0's from the 3-Fibonacci word: 1,2,1,1,2,1,2,1,1,2,1,2,1,1,2,1,...
From Michel Dekking, Oct 19 2016: (Start)
The sequence (a(n)) is fixed point of the morphism zeta given by zeta: 0->01, 1->02, 2->epsilon.
Here epsilon is the empty word. To see this, code the 0’s in the Fibonacci sequence followed by 0 by 5, and the 0’s followed by 1 by 6. Then add 2 after 5. This gives the morphism 1->52, 5->61, 6->61, 2->epsilon. Then injectivize, i.e., map 5 and 6 to 0.
The sequence (a(n)) is related to A108103. Let theta be the standard form of zeta: theta(1)=12, theta(2)=13, theta(3)=epsilon. Let psi be the morphism generating the version of A108103 with 2 and 3 interchanged, psi: 1->2, 2->131, 3->1. Then the unique fixed point of theta is different from the fixed points of psi, but theta and psi generate the same language, i.e., arbitrarily long words occurring in the fixed point of theta occur in the fixed points of psi. This is a nontrivial exercise (prove that 2 theta^{2n}(1) = psi^{2n}(2) 13 for all n>0).
The sequence (a(n)) is not related to A270788, which might be called the ternary Fibonacci sequence. The dynamical system generated by (a(n)) has an eigenvalue -1, whereas the system generated by A270788 is isomorphic to the Fibonacci dynamical system. (End)
The asymptotic density of the occurrences of 0, 1, and 2 is 1/2, 1/(2*phi) = A019827, and 1/(2*phi^2) = A187426 / 10, respectively, where phi is the golden ratio (A001622). The asymptotic mean of this sequence is (3-phi)/2 (A187798). - Amiram Eldar, May 28 2024

Paolo Xausa, Table of n, a(n) for n = 0..13529
Sam Coates, Metallic mean fractal systems and their tilings, arXiv:2405.04458 [cond-mat.other], 2024. See p. 13.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, arXiv preprint arXiv:1608.04487 [Math.DS], 2016.
Mahdi Saleh and Majid Jahangiri, On linear ternary Intersection sequences and their properties, arXiv:1709.03829 [math.CO], 2017. pages 5-8.

Cf. A003849, A108103.

Cf. A001622, A019827, A187426, A187798.

SubstitutionSystem[{0->{0,1}, 1->{0,2}, 2->{}}, {0}, {10}][[1]] (* Paolo Xausa, May 17 2024 *)

A247554 Decimal expansion of a(F_5), the maximum inradius of all triangles that lie in a regular pentagon of width 1.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 19 2014

			0.2440155280941711153813744336812161242644369887...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.19 Circumradius-Inradius Constants, p. 535.

Cf. A019827 (a(F_4)(unit square)).

a[F5] = Root[5*x^8 - 175*x^7 + 611*x^6 - 816*x^5 + 720*x^4 - 280*x^3 + 160*x^2 - 96*x + 16, x, 1]; RealDigits[a[F5], 10, 101] // First

solve(x=0, 1/4, 5*x^8 - 175*x^7 + 611*x^6 - 816*x^5 + 720*x^4 - 280*x^3 + 160*x^2 - 96*x + 16) \\ Michel Marcus, Sep 19 2014

Smallest positive root of the polynomial given in the Mathematica code.

cp's OEIS Frontend

A019830 Decimal expansion of sine of 21 degrees.

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A019907 Decimal expansion of tangent of 9 degrees.

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A340725 Decimal expansion of Gamma(9/10).

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A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

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A267860 An infinite ternary 3-Fibonacci sequence (replace each 00 factor of the Fibonacci word with 020).

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Mathematica

A247554 Decimal expansion of a(F_5), the maximum inradius of all triangles that lie in a regular pentagon of width 1.

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Mathematica

PARI

Formula