A019884 -id:A019884 - cp's OEIS frontend

A214726 Decimal expansion of the perimeter of Cairo and Prismatic tiles.

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

Offset: 1

Jonathan Vos Post, Jul 26 2012

An algebraic integer with degree 4 and minimal polynomial x^4 - 16x^2 + 16. - Charles R Greathouse IV, Apr 21 2016

Length of the longest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Dec 13 2020

			3.8637033051562731469989727989....

Paolo Xausa, Table of n, a(n) for n = 1..10000
Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner, Isoperimetric Pentagonal Tilings, Not. AMS, Vol. 59, No. 5 (May 2012) pp. 632-640; arXiv preprint, arXiv:1111.6161 [math.MG], 2011.

Cf. A002193, A002194, A019884.

First[RealDigits[Csc[Pi/12],10,100]] (* Paolo Xausa, Oct 19 2023 *)

2*sqrt(2+sqrt(3)) \\ Charles R Greathouse IV, Apr 21 2016

Equals 2*(sqrt(2+sqrt(3))).
Equals csc(Pi/12). - Amiram Eldar, May 28 2021
Equals sqrt(2) + sqrt(6). - Vaclav Kotesovec, May 28 2021
Equals Product_{k>=1} (25 - 144*k^2)/(100 - 144*k^2). - Antonio Graciá Llorente, Jul 13 2024
Equals 4 * A019884. - Alois P. Heinz, Jul 14 2024

a(100) corrected by Georg Fischer, Jul 12 2021

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

A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.9951847266721968862448369...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 16

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022

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

PARI
```
real(I^(1/16))
```
PARI
```
cos(Pi/32)
```
PARI
```
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).
Satisfies 32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +84480*x^8 -21504*x^6 +2688*x^4 -128*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

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A214726 Decimal expansion of the perimeter of Cairo and Prismatic tiles.

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

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A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

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