A134944 -id:A134944 - cp's OEIS frontend

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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, 8

Offset: 0

N. J. A. Sloane

Midsphere radius of regular icosahedron with unit edges.
Also half of the golden ratio (A001622). - Stanislav Sykora, Jan 30 2014
Andris Ambainis (see Aaronson link) observes that combining the results of Barak-Hardt-Haviv-Rao with Dinur-Steurer yields the maximal probability of winning n parallel repetitions of a classical CHSH game (see A201488) asymptotic to this constant to the power of n, an improvement on the naive probability of (3/4)^n. (All the random bits are received upfront but the players cannot communicate or share an entangled state.) - Charles R Greathouse IV, May 15 2014
This is the height h of the isosceles triangle in a regular pentagon, in length units of the circumscribing radius, formed by a side as base and two adjacent radii. h = sin(3*Pi/10) = cos(Pi/5) (radius 1 unit). - Wolfdieter Lang, Jan 08 2018
Also the limiting value(L) of "r" which is abscissa of the vertex of the parabola  F(n)*x^2 - F(n+1)*x + F(n + 2)(where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Feb 24 2021

			0.80901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Scott Aaronson, The NEW Ten Most Annoying Questions in Quantum Computing (2014).
Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, Rounding Parallel Repetitions of Unique Games (2008).
Irit Dinur and David Steurer, Analytical approach to parallel repetition, arXiv:1305.1979 [cs.CC], 2013-2014.
Michael Penn, a golden value of cosine., YouTube video, 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001611, A001622, A019827, A134972, A134944, A134946, A187426, A296182.

Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).

convert(sin(3*Pi/10),radical); # W. Edwin Clark, May 24 2023
Digits:=100; evalf((1+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[(1 + Sqrt[5])/4, 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[54 Degree],10,120][[1]] (* Harvey P. Dale, Apr 21 2018 *)

(1+sqrt(5))/4 \\ Charles R Greathouse IV, Jan 16 2012

Equals (1+sqrt(5))/4 = cos(Pi/5) = sin(3*Pi/10). - R. J. Mathar, Jun 18 2006
Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - R. J. Mathar, Oct 27 2008
Equals A001622 / 2. - Stanislav Sykora, Jan 30 2014
phi / 2 = (i^(2/5) + i^(-2/5)) / 2 = i^(2/5) - (sin(Pi/5))*i = i^(-2/5) + (sin(Pi/5))*i = i^(2/5) - (cos(3*Pi/10))*i = i^(-2/5) + (cos(3*Pi/10))*i. - Jaroslav Krizek, Feb 03 2014
Equals 1/A134972. - R. J. Mathar, Jan 17 2021
Equals 2*A019836*A019872. - R. J. Mathar, Jan 17 2021
Equals (A094214 + 1)/2 or 1/(2*A094214). - Burak Muslu, Feb 24 2021
Equals hypergeom([-2/5, -3/5], [6/5], -1) = hypergeom([-1/5, 3/5], [6/5], 1) = hypergeom([1/5, -3/5], [4/5], 1). - Peter Bala, Mar 04 2022
Equals Product_{k>=1} (1 - (-1)^k/A001611(k)). - Amiram Eldar, Nov 28 2024
Equals 2*A134944 = 3*A134946 = A187426-11/10 = A296182-1. - Hugo Pfoertner, Nov 28 2024
Equals A134945/4. Root of 4*x^2-2*x-1=0. - R. J. Mathar, Aug 29 2025

A374149 Decimal expansion of the minimum volume of an axis-aligned bounding box which includes the shortest minimum-link polygonal chain joining all the vertices of the cube {0,1}^3.

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Jun 29 2024

It has been proved that it is not possible to join the 8 vertices of a cube with a polygonal chain that has fewer than 6 edges (see Links, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, Theorem 2.2).
Here we are considering the additional constraint that asks to minimize the volume of the Axis-Aligned Bounding Box (AABB) including the above-mentioned optimal polygonal chain consisting of only 6 connected line segments and that joins all the vertices of the cube [0,1] X [0,1] X [0,1].
Given phi = (1+sqrt(5))/2, the well-known golden ratio (see A001622), a valid polygonal chain is (0, 1, 0)-(0, 0, 0)-((1+phi)/2, 0, (1+phi)/2)-(1/2, 1+phi, 1/2)-((1-phi)/2, 0, (1+phi)/2)-(1, 0, 0)-(1, 1, 0) (see Links, p. 164), and consequently the minimum volume AABB is [(1-phi)/2, (1+phi)/2] X [0, 1+phi] X [0, (1+phi)/2].
As noted by Hugo Pfoertner, the present sequence is also given by phi^5/2 (i.e., A244593/2).

			5.5450849718747371205114670859140952943...

Marco Ripà, General uncrossing covering paths inside the Axis-Aligned Bounding Box, Journal of Fundamental Mathematics and Applications, Volume 4, 2021, Number 2, Pages 154-166.

Cf. A001622, A134944, A225227, A244593, A261547, A363755, A373537.

RealDigits[(11+5*Sqrt[5])/4, 10, 100][[1]]

Equals phi*(1+phi)*((1+phi)/2), where phi := (1+sqrt(5))/2 is the golden ratio.
Equals (11+5*sqrt(5))/4.
Equals phi^5/2.
Equals 10*A134944 + 3/2.

A081009 a(n) = Fibonacci(4n+3) - 1, or Fibonacci(2n+2)*Lucas(2n+1).

Original entry on oeis.org

1, 12, 88, 609, 4180, 28656, 196417, 1346268, 9227464, 63245985, 433494436, 2971215072, 20365011073, 139583862444, 956722026040, 6557470319841, 44945570212852, 308061521170128, 2111485077978049, 14472334024676220, 99194853094755496

Offset: 0

R. K. Guy, Mar 01 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A134944.

List([0..30], n-> Fibonacci(4*n+3)-1); # G. C. Greubel, Jul 14 2019

[Fibonacci(4*n+3)-1: n in [0..30]]; // Vincenzo Librandi, Apr 15 2011

with(combinat) for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+3)-1) od # James Sellers, Mar 03 2003

Fibonacci[4*Range[0,30] +3] -1 (* G. C. Greubel, Jul 14 2019 *)
LinearRecurrence[{8,-8,1},{1,12,88},30] (* Harvey P. Dale, Sep 23 2019 *)

vector(30, n, n--; fibonacci(4*n+3)-1) \\ G. C. Greubel, Jul 14 2019

[fibonacci(4*n+3)-1 for n in (0..30)] # G. C. Greubel, Jul 14 2019

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (1+4*x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012
Product_{n>=1} (1 - 1/a(n)) = (5+sqrt(5))/8 = A134944 + 1/2. - Amiram Eldar, Nov 28 2024

More terms from James Sellers, Mar 03 2003

A375803 a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(2n-1) Fibonacci(2*n+1).

Original entry on oeis.org

0, 20, 195, 4420, 72624, 1347905, 23877840, 430583140, 7712000835, 138485573876, 2484341814240, 44584372180225, 800002107309600, 14355674602647860, 257600625681170499, 4622465972012379940, 82946715695078486160, 1488418904383171787585, 26708590219470770907120

Offset: 1

Amiram Eldar, Aug 29 2024

Amiram Eldar, Table of n, a(n) for n = 1..798
Hideyuki Ohtskua, proposer, Problem H-944, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 3 (2024), p. 266.
Index entries for linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1).

Cf. A000045, A059929, A064170, A134944, A374149, A375804.

a[n_] := Fibonacci[n-1] * Fibonacci[n+1] * Fibonacci[2*n-1] * Fibonacci[2*n+1]; Array[a, 20]

a(n) = fibonacci(n-1) * fibonacci(n+1) * fibonacci(2*n-1) * fibonacci(2*n+1);

a(n) = A059929(n-1) * A059929(2*n-1) = A059929(n-1) * A064170(n+2).
Sum_{n>=2} (-1)^n/a(n) = (5*sqrt(5) - 11)/4 = A374149 - 11/2 = 10 * A134944 - 4 (Ohtskua, 2024).
G.f.: -x^2*(-20+65*x+195*x^2-84*x^3-13*x^4+x^5) / ( (1+x)*(x^2-3*x+1)*(x^2-18*x+1)*(x^2+7*x+1) ). - R. J. Mathar, Aug 30 2024

A384682 Decimal expansion of (5/6)phi = 5(1 + sqrt(5))/12, where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, Jun 06 2025

			1.34836165729157904017...

Cf. A001622, A021016, A134944, A134946, A384238.

RealDigits[GoldenRatio * 5/6, 10, 100, 0][[1]]

Minimal polynomial: 36*x^2 - 30*x - 25.
Equals Integral_{x=0..1} sqrt(x + sqrt(x + sqrt(x + ...))) dx.
Equals Integral_{x=0..1} (1 + sqrt(1 + 4*x))/2 dx.
Equals 10*A134944/3 = 5*A134946. - Hugo Pfoertner, Jun 07 2025

A153506 Decimal expansion of (e+gamma+Pi+phi)/4, where gamma is the Euler-Mascheroni constant and phi is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 28 2008

			2.01378103392506654565850744477005148267922125220518796422...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A135000, A135001.

SetDefaultRealField(RealField(100)); R:= RealField(); phi:=(1+ Sqrt(5))/2; (Exp(1) + EulerGamma(R) + Pi(R) + phi)/4; // G. C. Greubel, Aug 31 2018

RealDigits[(E + EulerGamma + Pi + GoldenRatio)/4, 10, 50][[1]] (* G. C. Greubel, Aug 17 2016 *)

default(realprecision, 100); phi=(1+sqrt(5))/2; (exp(1) + Euler + Pi + phi)/4 \\ G. C. Greubel, Aug 31 2018

Equals A003881+A019741+A134944+A001620/4. - R. J. Mathar, Jan 25 2009

More digits from R. J. Mathar, Jan 22 2009

A290013 Length of the period of the continued fraction expansion of phi/n where phi is the golden ratio.

Original entry on oeis.org

1, 1, 2, 2, 1, 6, 2, 2, 6, 5, 4, 4, 1, 10, 8, 4, 3, 2, 8, 14, 2, 12, 10, 4, 11, 5, 14, 10, 4, 28, 8, 8, 8, 1, 20, 2, 7, 4, 8, 14, 6, 6, 18, 8, 24, 6, 2, 4, 22, 31, 12, 14, 9, 10, 2, 12, 16, 12, 20, 20, 5, 8, 8, 20, 13, 20, 22, 2, 10, 52, 28, 2, 15, 19, 36, 4

Offset: 1

Vanessa Gomez, Eric Jovinelly, Jacob A. McCann, Bryce Orloski, Catherine Rea, Shannon Talbott, Jul 17 2017

We calculated the continued fraction expansion of phi/n and observed that the expansion is periodic after the first nonzero term. We tracked the periodicity of the expansions and present them here. The authors acknowledge the National Science Foundation (DMS-1560019) and Muhlenberg College for supporting the REU (Research Experiences for Undergraduates) on which this sequence is based.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001622 (phi), A019863 (phi/2), A134943 (phi/3), A134944 (phi/4), A134946 (phi/6).

a[n_] := ContinuedFraction[GoldenRatio/n] // Last // Length; Array[a, 80] (* Jean-François Alcover, Jul 28 2017 *)

cp's OEIS Frontend

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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A374149 Decimal expansion of the minimum volume of an axis-aligned bounding box which includes the shortest minimum-link polygonal chain joining all the vertices of the cube {0,1}^3.

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A081009 a(n) = Fibonacci(4n+3) - 1, or Fibonacci(2n+2)*Lucas(2n+1).

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A375803 a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(2*n-1) * Fibonacci(2*n+1).

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A384682 Decimal expansion of (5/6)*phi = 5*(1 + sqrt(5))/12, where phi is the golden ratio.

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A153506 Decimal expansion of (e+gamma+Pi+phi)/4, where gamma is the Euler-Mascheroni constant and phi is the golden ratio.

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Magma

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A290013 Length of the period of the continued fraction expansion of phi/n where phi is the golden ratio.

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A375803 a(n) = Fibonacci(n-1) * Fibonacci(n+1) * Fibonacci(2n-1) Fibonacci(2*n+1).

A384682 Decimal expansion of (5/6)phi = 5(1 + sqrt(5))/12, where phi is the golden ratio.