A019863 -id:A019863 - cp's OEIS frontend

A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

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

Offset: 2

Jean-François Alcover, Jul 01 2014

Essentially the same digit sequence as A239798, A019863 and A019827. - R. J. Mathar, Jul 03 2014

The minimal polynomial of this constant is x^2 - 11*x - 1. - Joerg Arndt, Jan 01 2017

			11.09016994374947424102293417182819058860154589902881431067724311352630...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 83.

Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
Wikipedia, Hard Hexagon Model.
D. W. Wood and R. W. Turnbull, z^2-11z-1 as an algebraic invariant for the hard-hexagon model, 1988 J. Phys. A: Math. Gen. 21 L989.

Cf. A019863, A019827, A085850, A239798.

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458 A176522, A176537 (metallic means).

RealDigits[GoldenRatio^5, 10, 103] // First

(5*sqrt(5)+11)/2 \\ Charles R Greathouse IV, Aug 10 2016

Equals ((1 + sqrt(5))/2)^5 = (11 + 5*sqrt(5))/2.
Equals phi^5 = 11 + 1/phi^5 = 3 + 5*phi, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Nov 11 2023
Equals lim_{n->infinity} S(n, 5*(-1 + 2*phi))/ S(n-1, 5*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A296182 Decimal expansion of (2 + phi)/2, with the golden section phi from A001622.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jan 08 2018

In a regular pentagon this is the distance between a vertex and the midpoint of the opposite side in units of the radius of the circumscribing circle.

			1.809016994374947424102293417182819058860154589902881431067724311352630231409451...

Index entries for algebraic numbers, degree 2.

Cf. A001622, A019863, A081003, A322159.

RealDigits[(5 + Sqrt[5])/4, 10, 111][[1]] (* Robert G. Wilson v, Jan 14 2018 *)
RealDigits[(2+GoldenRatio)/2,10,120][[1]] (* Harvey P. Dale, Aug 10 2025 *)

(5 + sqrt(5))/4 \\ Michel Marcus, Jan 08 2018

Equals (2 + phi)/2 = (5 + sqrt(5))/4 = (2*phi - 1)*phi/2 = with phi from A001622.
Equals 1 + A019863.
From Amiram Eldar, Nov 28 2024: (Start)
Equals 1/A322159.
Equals Product_{k>=0} (1 + 1/A081003(k)). (End)

A204188 Decimal expansion of sqrt(5)/4.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Jan 14 2012

Equals Product_{n>=1} (1 - 1/A000032(2^n)).
Essentially the same as A019863 and A019827. - R. J. Mathar, Jan 16 2012
The value is the distance of the W point of the Wigner-Seitz cell of the body-centered cubic lattice (that is the Brioullin zone of the face-centered cubic lattice) to its four nearest neighbors. Let the points of the simple cubic lattice be at (1,0,0), (0,1,0), (1,0,0) etc and the point in the cube center at (1/2, 1/2, 1/2). Then W is at (0, 1/4, 1/2) [or any of the 24 symmetry related positions like (0, 3/4, 1/2), (0, 1/2, 1/4) etc.], and the four lattice points closest to W are at (-1/2, 1/2, 1/2), (0,0,0), (1/2, 1/2, 1/2) and (0,0,1). - R. J. Mathar, Aug 19 2013

			0.5590169943749474241022934171828190588601545899028814310677243113526302...

J. Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
Y. Tachiya, Transcendence of certain infinite products, J. Number Theory 125 (2007), 182-200.
Wikipedia, Brillouin zone

Cf. A001622, A002163.

Maple
```
evalf(sqrt(5)/4);
```

RealDigits[Sqrt[5]/4, 10, 100][[1]] (* Amiram Eldar, Dec 04 2018 *)

sqrt(5)/4 \\ Charles R Greathouse IV, Apr 21 2016

Equals sqrt(5)/4 = (-1 + 2*phi)/4, with the golden section phi from A001622.

Equals 5*A020837.

A340723 Decimal expansion of Gamma(3/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			2.991568987687590628312...

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

Cf. A019863, A340721, A340722, A340724.

Maple
```
evalf(GAMMA(3/10),120) ;
```

RealDigits[Gamma[3/10], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A340724 = Pi/A019863 [DLMF (5.5.3)]

this * A340722 * 2^(1/10)/sqrt(2*Pi) = A340721. [DLMF (5.5.5)]

A019866 Decimal expansion of sine of 57 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 05 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 16

RealDigits[Sin[57 Degree],10,120][[1]] (* Harvey P. Dale, Oct 30 2011 *)

sin(19/60*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals A019847 * A019880 + A019828 * A019861 = A019863 * A019896 + A019812 * A019845. - R. J. Mathar, Jan 27 2021

A049661 a(n) = (Fibonacci(6*n+1) - 1)/4.

Original entry on oeis.org

0, 3, 58, 1045, 18756, 336567, 6039454, 108373609, 1944685512, 34895965611, 626182695490, 11236392553213, 201628883262348, 3618083506169055, 64923874227780646, 1165011652593882577, 20905285872462105744

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..750 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Cf. A000045, A019863, A374149.

[(Fibonacci(6*n+1)-1)/4: n in [0..20] ]; // Vincenzo Librandi, Aug 23 2011

Table[(Fibonacci[6n+1]-1)/4,{n,0,20}] (* or *) LinearRecurrence[ {19,-19,1},{0,3,58},20] (* Harvey P. Dale, Aug 22 2011 *)

a(n)=fibonacci(6*n+1)>>2 \\ Charles R Greathouse IV, Aug 23 2011

From R. J. Mathar, Nov 04 2008: (Start)
G.f.: x*(3+x)/((1-x)*(1-18*x+x^2)).
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). (End)
a(n) = (-1/4+1/40*(9+4*sqrt(5))^(-n)*(5-sqrt(5)+(5+sqrt(5))*(9+4*sqrt(5))^(2*n))). - Colin Barker, Mar 03 2016
Product_{n>=1} (1 - 1/a(n)) = (sqrt(5)+3)/8 = phi^2/4 = cos(Pi/5)^2 = A019863^2 = (A374149 + 1)/10. - Amiram Eldar, Nov 28 2024

A019848 Decimal expansion of sine of 39 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This sequence is also decimal expansion of cosine of 51 degrees. - Mohammad K. Azarian, Jun 29 2013

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 05 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

Cf. A019812, A019822, A019835, A019845, A019886, A019873, A019896, A019863.

RealDigits[Sin[39 Degree],10,120][[1]] (* Harvey P. Dale, Oct 02 2015 *)

sin(13/60*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals A019835*A019886 + A019822*A019873 = A019845*A019896 + A019812*A019863. - R. J. Mathar, Jan 27 2021

A237129 Let d = d(1)d(2)... d(q) denote the decimal expansion of an angle d expressed in degrees. The sequence a(n) lists the angles such that sin(d) = cos(d(1)d(2)... *d(q)).

Original entry on oeis.org

90, 418, 450, 666, 726, 778, 786, 810, 1146, 1170, 1386, 1395, 1530, 1775, 1890, 2218, 2250, 2394, 2474, 2482, 2610, 2842, 2898, 2970, 3186, 3195, 3312, 3330, 3366, 3375, 3690, 3711, 3735, 3915, 3933, 3978, 4050, 4146, 4194, 4274, 4282, 4338, 4410, 4698, 4770

Offset: 1

Michel Lagneau, Feb 04 2014

			666 is in the sequence because sin(666°) = cos(6*6*6°) = -.8090169943749... = -phi/2 where phi is the golden ratio (1+sqrt(5))/2. (A019863)
418 is in the sequence because sin(418°) = cos(4*1*8°)= .84804809615... (A019867)
3915 is in the sequence because sin(3915°) = cos(3*9*1*5°)= -.70710678118654752440 = -1/sqrt(2). (A010503)

Michel Lagneau, Table of n, a(n) for n = 1..2000

with(numtheory):err:=1/10^10:Digits:=20:for n from 1 to 5000 do:x:=convert(n,base,10):n1:=nops(x):p:=product('x[i]', 'i'=1..n1):s1:=evalf(sin(n*Pi/180)):s2:=evalf(cos(p*Pi/180)):if abs(s1-s2)

A340724 Decimal expansion of Gamma(7/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.29805533264755778568...

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

Cf. A246745, A175380, A246745.

Maple
```
evalf(GAMMA(7/10),120) ;
```

RealDigits[Gamma[7/10],10,120][[1]] (* Harvey P. Dale, Mar 02 2023 *)

gamma(.7) \\ Charles R Greathouse IV, Nov 26 2024

this * A340723 = Pi/A019863 [DLMF (5.5.3)]

this * A175380 * 2^(9/10)/sqrt(2*Pi) = 2*A246745. [DLMF (5.5.5)]

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

cp's OEIS Frontend

A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

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A296182 Decimal expansion of (2 + phi)/2, with the golden section phi from A001622.

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A204188 Decimal expansion of sqrt(5)/4.

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A340723 Decimal expansion of Gamma(3/10).

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A019866 Decimal expansion of sine of 57 degrees.

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A049661 a(n) = (Fibonacci(6*n+1) - 1)/4.

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A019848 Decimal expansion of sine of 39 degrees.

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A237129 Let d = d(1)d(2)... d(q) denote the decimal expansion of an angle d expressed in degrees. The sequence a(n) lists the angles such that sin(d) = cos(d(1)d(2)... *d(q)).